The Weyl law for algebraic tori

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus $T$ with bounded analytic conductor. The analytic conductor which we use is defined via the local Langlands correspondence for tori by choosing a finite dimensional complex algebraic representation of the $L$-group of $T$. Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor.


Introduction
A basic question in the analytic theory of automorphic forms is the following: Question. Given a connected reductive algebraic group G over a number field k, how many irreducible cuspidal automorphic representations of G are there?
This work was supported by Swiss National Science Foundation grant PZ00P2 168164. 1 To make sense of the Question, one needs to choose a positive real-valued invariant by which to order the representations of G. Sarnak, Shin and Templier [SST16] have proposed using the analytic conductor.
On the groups GL m , the analytic conductor has a standard definition [IS00], but over more general reductive groups it is less well understood. The most canonical (but not necessarily the most practical) definition is through the local Langlands conjectures. Let r : L G Ñ GL m pCq be a finite dimensional algebraic representation of the complex L-group of G. The local Langlands conjectures predict the existence of maps r˚, v : A v pGq Ñ A v pGL m q at every place v of k from the local unitary dual A v of G to that of GL m . One then defines the analytic conductor cpπ, rq of an irreducible automorphic representation π with respect to r by where the conductors on the right hand side are the "classical" local analytic conductors on GL m . The universal counting Question seems to be quite difficult at the level of generality in which we have stated it. Only very recently has there been progress in a few special cases. Over an arbitrary number field, the cases G " GL 1 , GL 2 , and under additional assumptions, GL m with m ě 3 have been resolved in a preprint of Brumley and Milićević [BM18]. The case that G is a one-dimensional torus over Q splitting over an imaginary quadratic extension was treated in work of Brooks and the author [BP18], and Lesesvre has studied the case that G is the units group of a quaternion algebra in his Paris 13 PhD thesis [Les18].
In this paper, we present an answer to the Question for G " T a torus defined over a number field k and r an arbitrary complex algebraic representation of its L-group. Even though the groups we are dealing with are abelian, our results are not easy, as we work with a very general notion of conductor. Indeed, the difficulties involved are already evident in the intricacy of the statement of the final result. As its reward, working with such a general notion of conductor reveals some of the richness that any general answer to the Question must exhibit. For example, the power of X in the asymptotic count of automorphic characters (see (1.4)) need not be an integer, but rather is a positive rational with denominator at most m. Further, we find that arbitrary integer powers of log X are possible in the asymptotic count (see example 1.8). Another interesting aspect of our results is the resemblance of the automorphic counting question to the Manin conjecture, which we present in subsection 1.2.
We make some precise definitions in order to give the statement of our result. Let T be an algebraic torus defined over a number field k. Let ApT q denote the group of continuous unitary characters of T pkqzT pAq, where A is the adèle ring of k. We call elements of ApT q automorphic characters; they are the basic objects of study in this paper. Let K{k be the minimal Galois extension over which T splits, and let G " GalpK{kq. Let X˚pT q and X˚pT q be the character and cocharacter lattices of T , and p T " HompX˚pT q, Cˆq the complex dual torus. We make the identification X˚pT q " X˚p p T q. Each of these objects admits a natural action of G. Let L T " p T¸G be the L-group of T , and pick r : L T Ñ GL m pCq an algebraic representation of L T . Generally, we will write n " dim T and m " dim r. Pick ν a Haar measure on the locally compact group ApT q.
The main goal of this paper is to give an asymptotic formula for νptχ P ApT q : cpχ, rq ď Xuq, where cpχ, rq is the analytic conductor (defined in §2), as X tends to infinity. The statement of the result requires a few more constructions. The restriction of r to p T breaks up as a direct sum Let M be the set of co-weights µ appearing in this decomposition, counted with multiplicity. Let S Ď M denote a subset of the coweights with multiplicity and S c its complement. For such an S, we define the complex diagonalizable group The restriction r| p T is faithful if and only if Dp∅q " t1u, and in that case we let (1.4) A " ApT, rq " maxt dim DpSq`1 |S| : S Ď M, DpSq ‰ t1uu.
Theorem 1.1. Suppose that r| p T is faithful. Then there exists a non-zero polynomial P " P ν,r,T and δ " δ r,T ą 0 such that νptχ P ApT q : cpχ, rq ď Xuq " X A P plog Xq`O ν,r,T pX A´δ q.
If r| p T is not faithful, then the left hand side is infinite for some finite X. The dependence of P and the implicit constant on ν is linear, since Haar measure is unique up to scaling. Here is a simple corollary of Theorem 1.1.
Corollary 1.2. Let T be a torus of dimension n, r an m-dimensional complex representation of its L-group, and ν a Haar measure on ApT q. We have νptχ P ApT q : cpχ, rq ď Xuq " T,r,ν X n`1 m .
Proof. By Theorem 1.1 it suffices to give uniform lower and upper bounds on A. For the lower bound, note that DpM q " p T which gives A ě n`1 m . For the upper bound, observe that for any S Ď M we have dim DpSq ď |S|, since dim Dp∅q " 0 and codim ker µ ď 1 for any µ P M . Therefore A ď maxt |S|`1 |S| : S ‰ ∅u ď 2.
We can give an expression for the degree of the polynomial P , but this requires a few more definitions. Since M was formed from the restriction of a representation of L T , the group G acts on M , and also on the power set 2 M " tS : S Ď M u. This action preserves |S| as well as dim DpSq, so G also acts on the set (1.5) Σ " tS ‰ ∅ : dim DpSq`1 |S| " Au.
Since the deleted set is preserved by the action of r G, we also have that r G acts on r Σ 0 .
Theorems 1.1 and 1.3 settle a problem of Sarnak, Shin and Templier [SST16,(4)] for the universal family of automorphic characters on a torus in the greatest possible generality. Remarks: (1) If one is willing to assume the Lindelöf hypothesis for Hecke characters, then any δ ă p2m 2 q´1 is admissible in the statement of Theorem 1.1.
(2) If one assumes that r is irreducible, then λ " 1. The assumption that r is irreducible simplifies many examples, since it implies that r G " G and r Σ " Σ.
(3) One is also very interested in the form of the leading constant in the asymptotic in Theorem 1.1, especially if it admits an interpretation in terms of the geometry or arithmetic of T . While in principle our method yields an expression for the leading constant, it is not so easy to write down in explicit form. One reason is that at primes p that ramify in K{k, we can only show that the number of characters of T pk p q is not so large as to affect the power of X or log X in the final answer. Another reason is that we cannot exclude the possibility that terms corresponding to non-identity global units of T may contribute to the leading term of the polynomial P (see example 1.7). (4) The invariant A in Theorem 1.1 and the power of log X in Theorem 1.3 are sufficiently complicated as to suggest that any general answer to the Question at the beginning of this paper would be quite onerous to state in full generality. (5) To resolve the counting problem at archimedean places for general r, one seems to require the use of a Brascamp-Lieb inequality due to Barthe [Bar98]. This is striking, as one does not typically expect such a deep analytic input to be necessary to resolve such a question in analytic number theory. The use of the Brascamp-Lieb inequality suggests that the counting problem for a general reductive group is difficult indeed, as already in the case of tori one needs to go much beyond an explicit understanding of the local Langlands correspondence. (6) Another interpretation of the families of automorphic characters studied in this paper is the following. Let T , K{k be as above, T _ " HompX˚pT q, G m q be the algebraic dual torus, and S " Res K{k G m . Given a faithful irreducible algebraic representation r of L T , one obtains an injective morphism i : T _ Ñ S by restriction of r. Such an injective map i gives rise to an L-homomorphism L T Ñ L S, and so Langlands predicts that there exists a transfer of automorphic characters i˚: ApT {kq Ñ ApGL 1 {Kq. Conversely, given i : T _ Ñ S, there exists a faithful irreducible algebraic representation r of L T extending i such that Lps, χ, rq " Lps, i˚χq for all χ P ApT q, where the left hand side is the Langlands L-function and right hand side is the Hecke L-function. (7) The automorphic counting problem outlined at the beginning of this paper has applications to the Ramanujan conjecture on general reductive groups (see the surveys [Sar05] and [Sha04]). Outside the case G " GL n , the Ramanujan conjecture is known to be false, but all automorphic forms for which it fails are expected to arise as functorial transfers from lower rank groups. For analytic applications, one would like to show that the Ramanujan conjecture cannot fail "too often" in a quantitative sense in terms of analytic conductor. At the very least, to do so one would need to estimate the sizes of subfamilies of ApGq coming from functorial transfers of automorphic characters of tori, and the so present paper paves the way for putting the above program into action.
Example 1.4. Let T " GL 1 " G m . Then p T " Cˆand G is trivial. We choose r " id " z : CˆÑ Cˆas representation of the L-group. Then ApT q is the set of primitive Hecke characters over k, and cpχ, rq is the standard notion of analytic conductor of a Hecke character, which we denote by Cpχq in all of the examples that follow. The multiset of co-weights is the singleton M " tzu, and 2 tzu " t∅, tzuu. We have Dp∅q " t1u and Dptzuq " Cˆ, so we have A " 2, and deg P " 0. Therefore there are " c k X 2 primitive Hecke characters of analytic conductor bounded by X for some constant c k ą 0 with a power saving error term. The same result has also recently been announced by Brumley and Milićević.
Example 1.5. Let T " GL 1 " G m as above, but take as representation the 1001-dimensional representation r " z '1001 : CˆÑ GL 1001 pCq. The set ApT q consists of Hecke characters χ as above, whereas r assigns to χ the conductor Cpχq 1001 . The multiset of coweights is tz, . . . , zu, where z is repeated 1001 times, and only the full set has DpSq ‰ t1u. Thus, A " 2{1001, and one recovers that there are " c k,r X 2{1001 Hecke characters of r-conductor less than X. This shows that the power of X in Theorem 1.1 can be arbitrarily small.
Example 1.6. Keep T " GL 1 " G m as above, but take as representation r " z 2 ' z 3 : CˆÑ GL 2 pCq. This is a 2-dimensional faithful representation of the L-group but it is not irreducible. The set ApT q is as in the previous two examples, but now r assigns to χ the conductor Cpχ 2 qCpχ 3 q. The set of coweights is tz 2 , z 3 u, and the subsets S and groups DpSq are S " ∅, tz 2 u, tz 3 u, tz 2 , z 3 u, Dp∅q " t1u, Dptz 2 uq " µ 3 , Dptz 3 uq "˘1, Dptz 2 , z 3 uq " Cˆ.
Example 1.7. Let T " G mˆGm . This torus has L-group equal to CˆˆCˆ. There is no faithful irreducible representation of this L-group. Take the faithful 2-dimensional representation z 1 'z 2 . Classically, this corresponds to counting pairs of primitive Hecke characters pχ 1 , χ 2 q with the conductor Cpχ 1 qCpχ 2 q. There are " c 12 X 2 log X pairs of Hecke characters of conductor bounded by X, for some c 12 ą 0.
For a general torus T , we will see later that there is a term which potentially contributes to the main term of νptχ P ApT q : cpχ, rq ď Xuq as X Ñ 8 for each global unit of T . In the example T " G mˆGm here, there are four global units: p1, 1q, p1,´1q, p´1, 1q and p´1,´1q. It is interesting to note that the contribution from p1, 1q is of size X 2 log X, each of p1,´1q and p´1, 1q make a contribution of size X 2 , and p´1,´1q contributes a smaller a power of X.
Example 1.8. Let K{k be any degree n Galois extension with group G. Let T " pRes K{k G m q{G m , where G m is embedded diagonally. One can express the dual torus as p T " tpz 1 , . . . , z n q P`Cˆ˘n : z 1¨¨¨zn " 1u.
Define r : L T Ñ GL n pCq by setting rpp1,¨¨¨, 1q¸σq to be the permutation matrix in GL n pCq defined by σ P G Ď S n , and rppz 1 , . . . , z n q¸1q "¨z The set of coweights is tz i u i"1,...,n , where z i represents projection onto the i-th coordinate. For any ∅ ‰ S P 2 tz i u we have dim DpSq " |S|´1, so A " 1, and the maximum in (1.4) is attained for all S with |S| ě 2, and so r Σ 0 " tS : |S| ě 2u. Since r is irreducible we have r G " G. Suppose now that G » S n . Then r Gz r Σ 0 " tS : |S| " 2u, . . . , tS : |S| " nu ( and so we have νptχ P ApT q : c r pχq ď Xuq " c r,T Xplog Xq n´2 . This is an example of logarithms arising in the asymptotic formula for "natural" reasons, giving a negative answer to a question of Sarnak [Sar08]. Also note that if e.g. rK : ks " 4 and G fi S 4 the asymptotic is " c G Xplog Xq 3 for some constants c G , whereas if G » S 4 it is " c S 4 Xplog Xq 2 . This shows that the power of log X in the asymptotic formula is sensitive to the arithmetic of the torus.
1.2. Relation to the Manin conjecture. The automorphic counting Question introduced at the outset of this paper is reminiscent of the Manin conjecture on the number of rational points of bounded height on a Fano variety. We briefly review the latter to point out a few of its features.
Let V be a Fano variety over k, and L a very ample line bundle. Let s 0 , . . . , s m be global sections of L with no common zeros, and φ " φ L,s 0 ,...,sm : V Ñ P m be the natural morphism associated to these data. Let Hpxq be the absolute exponential Weil height on P m pkq. Then h φ pxq " hpφpxqq is a height function on V pkq relative to L, s 0 , . . . , s m . If s 1 0 , . . . , s 1 m is another choice of global sections for the same L with φ 1 " φ L,s 1 0 ,...,s 1 Let N 1 eff pV q be the closed cone of effective divisors. Conjecture 1.9 (Batyrev-Manin Conj. C 1 ). Let V be a Fano variety with canonical bundle ω V not effective. If U is sufficiently small, we have N U pL, Xq " cX αpLq plog Xq tpLq´1 as X Ñ 8 for some positive constant c. Here, αpLq " inftλ P R : λrLs`rω V s P N 1 eff u, and tpLq is the codimension of the minimal face of BN 1 eff containing αpLqrLs`rω V s. The analogy between the automorphic counting question and the Manin conjecture is as follows, and should be viewed as an expression of the deep conjectures of Langlands. The role of the ambient space is played by P m pkq Ø ApGL m q, into which V pkq Ø ApGq embeds. The embedding is given by the data L, s 0 , . . . , s m on the Manin side, and (conjecturally) on the automorphic side by r : L G Ñ GL m pCq. Indeed, L, s 0 , . . . , s m determine a morphism V Ñ P m whereas the representation r (conjecturally) determines r˚: ApGq Ñ ApGL m q. The absolute exponential Weil height Hpxq 6 for x P P m pkq on the Manin side corresponds to the analytic conductor cpπq, π P ApGL m q on the automorphic side. The height function h φ pxq relative to φ corresponds to the analytic conductor cpπ, rq relative to r as in (1.1).
The invariant αpLq appearing in the Manin conjecture and the invariant A appearing in Theorem 1.1 both are expressible in terms of combinatorial geometry problems, see the computations with matroids in section 6 of this paper.
At least in the special case of tori, Theorems 1.1 and 1.3 suggest that tpLq on the Manin side corresponds to the set of orbits r Gz r Σ 0 . In both cases, the power of log comes from the possible embeddings of V or ApT q in ambient space that are "extremal" in the combinatorial geometry problem defining αpLq or A.
The leading constant in Manin's conjecture has been given a conjectural interpretation in terms of adèlic volumes by Peyre [Pey95] and Chambert-Loir and Tschinkel [CLT10]. For a discussion of the significance of the leading constant in the automorphic counting problem, see [BM18,§1.5].
While the analogy presented here is striking, it only goes so far. In Manin's conjecture, there is a canonical choice of L, that is, one takes L "´ω V , the anti-canonical bundle. In the automorphic setting, there is apparently no canonical choice of complex representation r of the L-group of G. Moreover, in the setting of Manin's conjecture the set of possible height functions corresponds to the ample cone of V , whereas in the automorphic setting, the set of possible height functions on ApT q is a much larger object. Lastly, we remark that the invariant tpLq in Manin's conjecture is an essentially global invariant of V . On the other hand, r Gz r Σ 0 has a somewhat more local nature, as we shall see in section 7 of this paper.
1.3. Outline of the proof. To prove Theorems 1.1 and 1.3, it suffices to study the generating series Zpsq " ż ApT q 1 cpχ, rq s dνpχq.
In this paper we show that Zpsq converges absolutely and uniformly on compacta in the right half-plane Repsq ą A. Then we prove that Zpsq continues to a meromorphic function polynomially bounded in vertical strips in the right half-plane Repsq ą A´p2m 2 q´1, and show that the only pole of Zpsq in this domain occurs at s " A and that it is of order | r Gz r Σ 0 |. Theorems 1.1 and 1.3 then follow from a standard application of Perron's formula.
In section 2 we study the local analytic conductors in detail, taking particular care to deal with places ramified in K{k. We define "abelian" local analytic conductorsc v pχ, rq for each nonarchimedean place v of k that control the standard local analytic conductors, but which are accessible to the counting techniques later in the paper. We also restrict the local Langlands correspondence to a distinguished compact subgroup of T pk v q at each finite place v of k.
The main goal of section 3 is to decompose Zpsq into a product of local generating series at each place v of k. This is not immediately possible due to the presence of the global units of T . What is possible is to express Zpsq as a sum over the global units of series attached to each unit, which themselves factor over places of k. We also prove a key global finiteness lemma related to the units of T in section 3, and the second assertion of Theorem 1.1.
Sections 4, 5 and 6 are devoted to the computation of the local generating series at unramified, ramified and archimedean places, respectively. The heart of the paper is in section 4, where an expression for the Dirichlet series coefficients at unramified primes is given in terms of the dimensions of the groups DpSq, and the number of fixed points of local Galois group actions, already reminiscent of those in the statement of Theorem 1.3. In section 5 we cannot give such an explicit evaluation of the local generating series at ramified primes, but we can show that these series converge absolutely in a right half-plane containing s " A. Since there are only finitely many ramified primes, this is sufficient for our purposes but obscures the leading constant of the polynomial P in Theorem 1.1. In section 6 we study the archimedean local generating series. To show that these converge absolutely in a right half-plane containing s " A, we make use of a Brascamp-Lieb inequality and the theory of matroids.
Finally, in section 7 we pull together the local results of sections 4, 5 and 6 to show that the global generating series attached to each global unit admits an analytic continuation by comparison with a product of Hecke L-functions. The global finiteness lemma of section 3 shows that as we vary the global unit, only finitely many different such L-series can occur. The sum over global units then converges absolutely, establishing the analytic properties of Zpsq.

Notation
Definition Location π F , π L , p, P uniformizers of F , L, and the maximal ideals they generate §2.2 p H n Tate cohomology groups §2.2 R the restriction to OL map out of H 1 pW L{F , p T q (2.10) T, n 1 , n 2 , n 3 T pF q » T " pRˆq n 1ˆp S 1 q n 2ˆp Cˆq n 3 , F local non-arch (2.20) T^T^" piR n 1ˆp Z{2Zq n 1 qˆZ n 2ˆp iR n 3ˆZ n 3 q (2.22) ppw, ǫq, α, pw 1 , α 1 qq a typical element of T^ §2.4 pa, c, pb, b 1 qq an element of X˚pG m q n 1ˆX˚p S 1 q n 2ˆX˚p Res C{R G m q n 3 §2.4 matrices formed from explicit parameterization of coweights §2.4 local generating series at unramified places (3.8) R v ps, xq, A v ps, xq similar, at ramified and archimedean places (3.9),(3.10) B set of places of k with pq kv , λq ‰ 1 or v ramified §3.1 T S an auxiliary torus attached to S P 2 M §3.2 α´1pxq fibers of a canonical map α : T S Ñ T §3.2 K 1 , Γ a field s.t. all components of α´1pxq are defined, GalpK 1 {kq §3.2 ppV q the set of geometric components of a variety V §3.2 N the non-negative integers §4 P ď pcq a finite subgroup of Hom G pOL , p T q (4.2) P " pcq a "sharp" subset of P ď pcq (4.3) Π ď pc, xq, Π " pc, xq character sums over P ď pcq and P " pcq (4.4) D k pcq a generalization of DpSq (4.7) f, ℓ, charpℓq the residue fields of F, L, their characteristic §4 apS, xq the number of Frobenius-fixed components of α´1pxq (4.11) apSq apS, 1q; the number of Frobenius-fixed points of π 0 pDpSqq (4.12) S red the maximal Galois-stable subset of S (4.14) H M a polytope in R m ě0 given by a matrix M (6.5), (6.6) B 8 inft}x} 8 : x P H M u (6.7) pN, Iq, rpSq, a matroid, its rank function Def 6.6, 6.8 P I , P B the matroid polytope and matroid base polytope Def 6.9 P 1 , D P , D P 1 a prime of K 1 above P, decomposition groups of P, P 1 §7.1 « equal up to an absolutely convergent Euler product Def 7.2 C a conjugacy class of Γ (7.7) Σ P subset of Σ fixed by D P (7.8) a C pS, xq the number of C-fixed components of α´1pxq (7.10) Σ a,b set of S P 2 M such that dim DpSq " a and |S| " b (7.12) r Σ a,b similar to r Σ 0 , but with respect to Σ a,b (7.14) V a,b , ψ a,b permutation representation of Γ on r Σ a,b , its character §7.1 1.5. Acknowledgements. This paper was began as joint work with Hunter Brooks, who, soon after it was started, left academic mathematics and went into industry. I learned a great deal of mathematics from Hunter, especially the background for section 2 of this paper. I would also like to express my gratitude to Will Sawin. This paper grew out of discussions of Hunter, Will and I in Zürich in fall 2016, and would not have been possible without Will's generosity with his ideas. Lastly, thanks are due to Rico Zenklusen, who pointed out to me that the polytope minimization problem in section 6 was best studied in the language of matroids, and introduced me to the polymatroid intersection theorem which plays a key role in that section. I would also like to thank Farrell Brumley for interesting discussions on the automorphic counting problem, and Emmanuel Kowalski and Peter Sarnak for their encouragement. I would like to thank the Swiss National Science Foundation for its financial support, and the people of Switzerland for their commitment to funding fundamental research.

Conductors
Let v be a place of k and denote by k v the completion of k at v. An automorphic character χ P ApT q admits a factorization in which all but finitely many of the χ v are trivial on the maximal compact subgroup of T pk v q. Likewise, the global L-group L T gives rise to local L-groups at each place of T . Indeed, the torus T is defined over k v and for each v there exists a unique minimal Galois extension K w of k v splitting T . We have GalpK w {k v q ãÑ GalpK{kq, and local L-groups L T v ãÑ L T , and so r determines a finite dimensional complex algebraic representation of the local L-group L T v for each v by restriction. The analytic conductor cpχ, rq is a local invariant of ApT q defined by where c v pχ v , r|L Tv q are local analytic conductors, all but finitely many of which are equal to 1. Thus, it suffices to restrict our attention to local analytic conductors, which are the subject of the rest of this section of the paper. For the remainder of section 2, we let T denote a torus defined over a local field F , splitting over a minimal Galois extension L{F . We now write G " GalpL{F q for the local Galois group and L T " p T¸G for the local L-group, dropping the subscript v. Finally, we suppose we are given a finite dimensional complex algebraic representation r : L T Ñ GLpV q, and write cpχ, rq for the local analytic conductor (again dropping v from the notation).
Following [Bor79], we define cpχ, rq by passing through the local Langlands correspondence and taking the conductors from the Galois representation associated to χ and r. To that end, recall the Weil group W F of a local field (see e.g. [Tat79,§1]), and the relative Weil group The relative Weil group can be thought of as a group extension of G by Lˆ, i.e. the sequence is exact, where the first map is the Artin map of local class field theory. For a topological abelian group A, HompA, Cˆq denotes the group of continuous complex characters of A, and A^denotes the subgroup of unitary characters, that is to say the Pontryagin dual. For M a G-module, H 1 pG, M q denotes the first group cohomology group defined using continuous cocycles. The local Langlands correspondence for tori [Lan97] asserts that there is a canonical isomorphism where W L{F acts on p T via the map σ (see (2.2)). Langlands also shows that (2.3) restricts to where p T u " X˚pT q^" HompX˚pT q, S 1 q. We will use (2.4) when F is an archimedean local field. If χ P HompT pF q, Cˆq corresponds to ξ P H 1 pW L{F , p T q across (2.3) and σ is as in (2.2), then is a conjugacy class of homomorphisms called the Langlands parameter of χ. The composition r˝ϕ : W L{F Ñ GLpV q is the complex Galois representation associated to χ and r. One defines local L and ε-factors associated to finite dimensional complex Galois representations as in [Tat79,§3]. If F is nonarchimedean, we define local analytic conductor cpχ, rq in terms of the ε-factor of r˝ϕ, and if F is archimedean we define cpχ, rq in terms of the L-factor of r˝ϕ.
We now summarize the contents of the rest of section 2. In subsections 2.1-2.3 we restrict to the non-archimedean case. In subsection 2.1, the ε-factor of a complex Galois representation p̺, V q is given in terms of the Artin conductor cp̺q of p̺, V q (see (2.6)). If L{F is ramified we are unable to perform the counting arguments in section 4 with respect to the Artin conductor. To get around this obstacle we introduce the abelian conductorcp̺q (see Definition 2.5), which is more amenable to counting (see section 5) and which controls the Artin conductor (see Lemma 2.7).
The goal of subsections 2.2 and 2.3 is to restrict the Langlands isomorphism (2.3) to the ring of integers of F . It turns out that it is more natural to restrict to the image of the norm map N : T pO L q Ñ T pO F q. In subsection 2.2 we prove the key lemma that the image of the norm map has finite index. In subsection 2.3 we accomplish the restriction.
In subsection 2.4 we study the archimedean case.
2.1. The Artin conductor. We now restrict our attention to non-archimedean local fields. Let dx be a Haar measure on F , ψ a non-trivial additive character of F , and dx 1 the dual Haar measure relative to ψ. Given a finite dimensional complex representation p̺, V q of the Weil group W F , Tate [Tat79,§3] defines the ε-factor εpV, ψ, dxq " εp̺, ψ, dxq attached to these data.
Tate [Tat79,§3.4.2] shows that εpV, ψ, dxq is additive, and in particular only depends on the isomorphism class of V . If p̺, V q is unitary we have (see [Tat79,§3.4.7]) that (2.6) |εpV, ψ, dxq| 2 " q cp̺q F pδpψqdx{dx 1 q dimpV q . In particular, since εpV, ψ, dxq only depends on the isomorphism class of p̺, V q, it suffices for (2.6) to hold that p̺, V q be unitarizable. Here q F is the cardinality of the residue field of F and cp̺q is the Artin conductor of the representation p̺, V q. The factor pδpψqdx{dx 1 q dimpV q is explained in Tate and we do not need to elaborate on it here.
Next, we review the definition of the Artin conductor cp̺q of a finite-dimensional complex representation ̺ : W F Ñ GLpV q of the Weil group of a non-archimedean local field. The classical Artin conductor is an invariant of a finite dimensional complex representation of a finite Galois group GalpE{F q. For more discussion of the classical Artin conductor see [Ser79,Ch.VI] or [Ulm16,§4]. We give a slightly nonstandard definition of the Artin conductor of a finite-dimensional complex representation of W L{F following [Ulm16] (this goes back at least to [DDT97]).
Let G F " GalpF {F q and let W F ãÑ G F be the inclusion given as part of the data of a Weil group (see [Tat79,§1.4.1]). For any v P r´1, 8q, let W v F be the inverse image of the (upper numbering) higher ramification group G v F by this inclusion (see Serre [Ser79] for definitions, especially Ch. IV, §3, Remark 1). Let W v L{F be the image of W v F by the canonical projection W F ։ W L{F . We have therefore a decreasing filtration of groups:¨¨Ď is called the Artin conductor of p̺, V q. The value of cp̺q only depends on ̺| W 0 L{F , and extends the notion of Artin conductor for complex representations of finite Galois groups.
Proof. Since there are no breaks in the upper-numbering filtration between´1 and 0, and the upper numbering is left-continuous (see [Ser79,Ch.4 §3]), it follows that the Artin conductor cp̺q only depends on the restriction of ̺ to W 0 L{F . Since W 0 L{F is compact and profinite, and GLpV q has no small subgroups, it follows that H " ker Therefore to ̺| W 0 L{F there is associated a finite extension E{F with E Ď L ab , and ̺ factors through the representation ̺ 1 : GalpE{F q Ñ GLpV q given by composing with the isomorphisms above. It is shown in [Ulm16,§4] for finite dimensional complex representations of finite Galois groups that the standard definition of the Artin conductor matches the one given in the Proposition/Definition with the higher ramification groups G v pE{F q in place of the Weil group and ̺ 1 in place of ̺.
The Artin conductor is a special case of the following more general notion of conductor.
Definition 2.3. For G a group endowed with a filtration F " pG v q and ̺ : G Ñ GLpV q a finite dimensional complex representation, we call With G " W L{F and F given by the upper numbered filtration, c F p̺q the Artin conductor of p̺, V q.
Next, we introduce an "abelian" conductorcp̺q. The Artin conductor of a representation p̺, V q is controlled by the abelian conductor, and in the case that the representation factors through W L{F for L{F an unramified extension, the abelian conductor is identical to the Artin conductor.
We denote for any v P p´1, 8q the groups O Lemma 2.4. For any real number v ą´1, the following diagram commutes and the horizontal rows are short exact: We apply [Ser79, Ch. IV, Prop. 14] with G " W L{F and H " W ab L to see that Finally, by the third group isomorphism theorem and [Tat79, §1.1] we conclude that We are finally ready to give a definition of the abelian conductorcp̺q.
Definition 2.5. Let p̺, V q be a finite-dimensional complex representation of W L{F , W 0 L{F , Lˆ, or OL , where the latter two groups are viewed as subgroups of W L{F as in Lemma 2.4. Let U be the filtration given by pO pvq L q. Thenc p̺q " c U p̺q is called the abelian conductor of p̺, V q.
Finally, we note that the abelian conductor controls the Artin conductor and vice-versa. Let v 0 " inftv : G v pL{F q " t1uu. For example, if L{F is unramified then v 0 "´1.
Proof. For the first inequality, simply observe that For the second, we use the fact that if v ą v 0 then codimpV ̺pO pvq L q q " codimpV ̺pW v L{F q q. We have cp̺q " Corollary 2.8. We have for r, χ, ξ, ψ, dx as above that The reason we prefer the abelian conductor is the following.
Proposition 2.9. Let r be a finite-dimensional complex algebraic representation of L T of dimension m. Let M be the set of co-weights of r with multiplicity, as in (1.2). Suppose that χ Ø ξ under the local Langlands correspondence (2.3). Theñ where c is the standard notion of conductor of a quasicharacter of Lˆ.
Proof. Let ϕ " ξˆσ : W L{F Ñ L T be the Langlands parameter associated to χ P HompT pF q, Cˆq by the local Langlands correspondence. By definition, the abelian local analytic conductor only depends on We have that so that by the additivity of the abelian conductor cpr˝ϕq " ÿ µPM cpµ˝ξq.
14 2.2. The norm map. Because of the demands of our local to global decomposition in section 3, we must restrict the local Langlands correspondence (2.3) to a compact subgroup of T pF q which is finite index in the maximal compact subgroup T pO F q. As we will see in the following subsection 2.3, the natural choice is to restrict the Langlands correspondence to the image of the norm map (2.7) N : T pO L q Ñ T pO F q defined by the product of Galois conjugates. In this subsection prove an important lemma describing the image of N , which we write N T pO L q. We begin with a preliminary but crucial result.
More generally, we have the following result.
Lemma 2.11. We haveˇˇˇˇT Proof. We have an exact sequence and since X˚pT q is a free abelian group this leads to Recall that (see [Ono61, §2.1]) T pO F q » Hom G pX˚pT q, OL q, and that this is the unique maximal compact subgroup of T pF q » Hom G pX˚pT q, Lˆq. Taking G-invariants we have The rightmost map N above is given by where ℓ σ pχq " ℓpχ σ´1 q. Let K " kerpN : HompX˚pT q, Zq Ñ HompX˚pT q, Zqq " tℓ P HompX˚pT q, Zq :

Now
T pLq " HompX˚pT q, Lˆq and T pF q " Hom G pX˚pT q, Lˆq and the middle map N is given by where ψ σ is given by ψ σ pχq " ψpχ σ´1 q σ . Let Define the valuation map v : The snake lemma gives us an exact sequence Indeed, let ℓ P K be arbitrary. We claim that e L{F ℓ P vpT 1 q. The first claim follows from this second claim on letting ℓ run through a Z-basis for K, so it suffices to show this. Now we show the second claim. Choose π F a uniformizer for F . For ℓ P K let since ℓ P K. Therefore ψ ℓ P T 1 . Note also that vpψ ℓ qpχq " vpψpχqq " vpπ ℓpχq F q " e L{F¨ℓ pχq. Thus we have shown that for all ℓ P K we have e L{F ℓ P vpT 1 q, as claimed.
By the exact sequence (2.8) we have thaťˇˇˇT Recall the definition of the Tate which we call the Langlands pairing. We write G 0 for the inertia subgroup of G. The following is the main result of this section.
Proposition 2.12. Write RpH 1 q for the image of the restriction to OL map The subgroup RpH 1 q of Hom G pOL , p T q is of index at most ď |H 1 pG 0 , p T q|¨|H 2 pG 0 , p T q|. The Langlands pairing restricts to a perfect pairing In particular, if L{F is unramified, Langlands pairing restricts to a perfect pairing Corollary 2.13. The abelian local analytic conductorcpχ, rq only depends on χ| N T pO L q .
Of course, we would like the group RpH 1 q to be of finite index, and indeed this is true.
Lemma 2.14. We have that H i pG 0 , p T q is a finite group for all i ě 1.
Lemma 2.14 follows from a result of Cartan and Eilenberg, which we recall now since it will also be useful for other purposes later.
Theorem 2.15 (Duality Theorem). Let G be a finite group, A a G-module and C a divisible abelian group. For any i P Z there exists a perfect pairing Proof. See [CE56] chapter XII, theorems 4.1 and 6.4.
Proof of Lemma 2.14. In Theorem 2.15 we take G " G 0 , A " p T , and C " Cˆto obtain that s a perfect pairing. Now, by [AW67, §6 Cor. 1] we have that H i pG 0 , p T q is a group of finite exponent. Furthermore, X˚pT q is a finitely-generated G 0 -module, so by [AW67, §6 Cor. 2] we have that H i pG 0 , X˚pT qq is a finite group. The result now follows from the duality theorem.
To prepare for the proof of Proposition 2.12, we review the proof of the Langlands correspondence (2.3). To do so, we recall the following explicit descriptions of group cohomology and homology (the same exposition appeared in the appendix of [BP18]).
For this paragraph, let G be a group and M a left G-module. Computing via the inhomogeneous resolution gives the usual description of group cohomology tmaps ξ : G Ñ M satisfying ξpghq " ξpgq`gξphqu tmaps such that there exists m P M, ξpgq " gm´mu .
If ' S N is a direct sum of copies of an abelian group N indexed by a set S, let δ s pnq P ' S N be the element which is n in the spot indexed by s and 0 elsewhere. Computing via the inhomogeneous resolution then gives the following description of group homology where dpδ g,h pmqq " δ h pg´1mq´δ gh pmq`δ g pmq. If G 1 ă G is a finite-index normal subgroup, there is an action of G{G 1 on H 1 pG 1 , M q by the rule g˚δ g 1 pmq " δ gg 1 g´1 pgmq. There also exists a natural map which may be computed as follows: pick coset representatives g 1 , g 2 , . . . , g n for G{G 1 . Then any g P G determines a permutation τ P S n by the rule g i g " g 1 g τ piq (where g 1 P G 1 ), and Now we return to our review of the Langlands correspondence (2.3). In particular, G " GalpL{F q again. Let us begin by recalling two easy steps. First, we have the following standard isomorphisms and taking G-invariants Second, Langlands proves the following mild extension of Theorem 2.15. Let α P W L{F , χ P X˚pT q such that δ α pχq is a cycle representing a class in H 1 pW L{F , X˚pT qq. Let ξ be a cocycle representing a class in H 1 pW L{F , p T q. Langlands [Lan97] shows that the pairing Here N is the norm map defined as a product of Galois conjugates, and the vertical map is the trace map (2.16). Composing with the isomorphisms (2.13) and (2.14), we have that the norm map N : T pLq Ñ T pF q factors through the homology group H 1 pW L{F , X˚pT qq: be the annihilator of N T pO L q with respect to the Langlands pairing (2.9). To prove the proposition, it suffices to compute AnnpN T pO L qq, and show that is as described in the statement of the proposition.
Lemma 2.16. Let be the restriction to OL map (note OL acts trivially on p T ). Then we have Proof. Let t : H 1 pW L{F , X˚pT qq Ñ T pF q be the isomorphism obtained by composing the trace map (2.16) with the series of isomorphisms (2.14). The first step is to give an explicit description for the inverse image t´1pN T pO L qq ď H 1 pW L{F , X˚pT qq. Later, we use the explicit description for the cup product pairing (2.15) to compute AnnpN T pO L qq.
The main trick to compute t´1pN T pO L qq is to use the commuting triangle (2.17), as the trace map is difficult to work with directly. Restricting (2.17) to the maximal compact of T pLq we obtain where all arrows are surjective. Since we understand the diagonal arrow much better than the vertical one, this yields a description for t´1pN T pO L qq. In the above explicit description for group homology, it is the subgroup of H 1 pW L{F , X˚pT qq generated by sums of all possible homology classes δ α pχq as α runs over α P OL Ă W L{F .
We now use the description t´1pN T pO L qq " xδ α pχqy αPOL and compute the annihilator First we prove AnnpN T pO L qq Ě kerpRq. Let ξ represent a class in kerpRq. Then ξ vanishes on OL by definition, and we have ξ Y δ α pχq " 1 for any δ α pχq P t´1pN T pO L qq by our explicit description in the previous paragraph, and the definition of (2.15). Therefore kerpRq Ď Annpt´1pN T pO L qqq. Now we prove AnnpN T pO L qq Ď kerpRq. Suppose ξ P H 1 pW L{F , p T q does not represent any class in kerpRq. Then there exists a β P OL for which ξpβq ‰ 1. Since ξpβq ‰ 1 there exists χ P X˚pT q not vanishing on ξpβq P p T . Since OL acts trivially on X˚pT q, we have that χ β is a cycle, and thus represents a homology class. Thus δ β pχq P t´1pN T pO L qq and ξ Y δ β pχq ‰ 1, so ξ R Annpt´1pN T pO L qqq. Therefore kerpRq c Ď Annpt´1pN T pO L qqq c , so we have kerpRq " Annpt´1pN T pO L qq.
By Lemma 2.16 we have shown that s a perfect pairing. It now suffices to show that is of index at most ď |H 1 pG 0 , p T q|¨|H 2 pG 0 , p T q|, as in the statement of Proposition 2.12. Consider the inertia group W 0 L{F acting on p T , and the exact sequence as in Lemma 2.4. We take the inflation-restriction exact sequence attached to these data These give We take Frobenius invariants of this to obtain a sequence where 1 P Z acts by arithmetic Frobenius on p T . Since a cocycle is determined by its value on a generator, we have Therefore r 1 pH 1 pW 0 L{F , p T q Z q has index at most |H 1 pG 0 , p T q| in kerpgq Z , and kerpgq has index at Consider again W L{F acting on p T , and take the exact sequence Taking the inflation-restriction exact sequence associated to these we have Here the term H 2 pZ, p T G 0 q vanishes because the cohomological dimension of Z is one (see [Bro94, Ch. VIII, §2]). We have that RpH 1 q " pr 1˝r2 qpH 1 pW L{F , p T qq, and by the above remarks we conclude that RpH 1 q has index at most |H 1 pG 0 , p T q|¨|H 2 pG 0 , p T q| in Hom G pOL , p T q, as was to be shown.
In the case that L{F is an unramified extension, we have G 0 " t1u, so that the first part of Proposition 2.12 gives us that RpH 1 q " Hom G pOL , p T q. We have N T pO L q " T pO F q by Lemma 2.10, so that the Langlands pairing restricts to the perfect pairing (2.11). 20 2.4. Archimedean local fields. We assume in this subsection that F, L are archimedean local fields, with T defined over F and splitting over L. Let Γ R psq " π´s {2 Γps{2q and Γ C psq " 2p2πq´sΓpsq. If p̺, V q is a complex Galois representation of the Weil group W F , then the L factor of p̺, V q is of the form Let dx be a Haar measure on F , ψ a non-trivial additive character of F , and dx 1 the dual Haar measure relative to ψ.
Definition 2.17. Suppose F is an archimedean local field. If ξ corresponds to χ P HompT pF q, Cˆq under the Langlands isomorphism (2.3), σ is as in (2.2), and ϕ " ξˆσ, then the quantity is called the archimedean local analytic conductor of χ with respect to r.
Note that our definition differs slightly from the standard definition in that the`1 above is typically replaced by a`2 or`3. The reason some authors prefer`2 or`3 is to ensure that as dim V varies, there are only a finite number of representations of bounded conductor. Since we will always consider dim r to be fixed in this paper, we prefer`1 as it makes some of the computations in section 6 more elegant.
The definition of the L-factor Lps, V q for archimedean places is given in [Tat79] sections 3.1.1, 3.1.2 and 3.3.1 in terms of the classification of the finite dimensional irreducible representations of W F given in section 2.2.2. Therefore, we must make explicit parameterizations of the possible Langlands parameters ϕ " ξˆσ : W L{F Ñ L T , as well as the possible representations r : L T Ñ GLpV q, in order to be able to find their compositions among the classification [Tat79, §2.2.2].
For the rest of this subsection we assume T is defined over an archimedean local field F » R. The case that F » C is strictly easier, and follows closely the case that T » pRes C{R G m q n , which we elaborate on below.
A torus defined over F splits over a quadratic extension L » C, and so G " GalpL{F q is a group of order two, whose elements we write t1, τ u. Let us recall the explicit description of the Weil groups for archimedean local fields. We have W F " Lˆ\ Lˆj, and W L " Lˆ, where we write elements of W F as words in z and j and we have the rules jzj´1 " τ z and j 2 "´1. We also have π : W F {F " W ab F » Fŵ here the isomorphism π is given by πpzq " |z| 2 , and πpjq "´1.
In this subsection, we take L T " p T¸G to be the local archimedean L-group, and L T u " p T u¸G , where p T u " HompX˚pT q, S 1 q. The reason to study L T u instead of L T is that the unitary characters of T pF q correspond under the local Langlands correspondence (2.3) to H 1 pW L{F , p T u q (see [Lan97, Thm. 1]). Now we choose isomorphisms L » C and (2.20) T pF q » T " pRˆq n 1ˆp S 1 q n 2ˆp Cˆq n 3 so that dim T " n " n 1`n2`2 n 3 . By computing with the inflation-restriction exact sequence and using facts about the group cohomology of finite cyclic groups, we have an explicit parameterization of the L-equivalence classes of Langlands parameters ϕ : W L{F Ñ L T u . They are given by and ϕpjq "´p´1q ǫ 1 , . . . , p´1q ǫn 1 , 1 . . . , 1, p1, p´1q α 1 1 q, . . . , p1, p´1q α 1 n 3 q¯¸τ.
Here, w i P iR, ǫ i P t0, 1u, α i P Z, α 1 i P Z, and w 1 i P C such that w 1 i´α 1 i P iR. We write (2.22) T^" piR n 1ˆp Z{2Zq n 1 qˆZ n 2ˆp iR n 3ˆZ n 3 q, so that Langlands parameters may be parameterized by ppw, ǫq, α, pw 1 , α 1 qq P T^.
We also need an explicit description of the representation r. The representation r decomposes into irreducible representations, and we can parameterize all irreducible representations of L T by the set of orbits GzX˚p p T q using Mackey theory (see [Ser77, §8.2]). We now study this parameterization explicitly. Corresponding to (2.20) we have an isomorphism of G-modules where X˚pG m q " Z with G acting trivially, X˚pS 1 q " Z with τ P G acting by sending´1 to 1, and X˚pRes C{R G m q " Z 2 with τ P G acting by swapping the two factors. Each x P X is contained in a G-orbit of size 1 or 2. We have the following three genres of isomorphism classes of irreducible representations of L T : 1a) If x is fixed by G, i.e. is in an orbit of size one, then µ x is an irreducible representation of L T . 1b) If x is fixed by G, i.e. is in an orbit of size one, then µ x bpsignq is an irreducible representation of L T . 2) If x is not fixed by G, i.e. is in an orbit of size two, then V x " Ind L T p T µ x is an irreducible representation of dimension two of L T . It only depends on the orbit of x. That is, V x » V τ x and this representation is not isomorphic to any other V x 1 , x 1 ‰ x, τ x.
Therefore we get a decomposition for some x i , x 1 i which are fixed by G and some x 2 i which are not fixed by G, and where m 1`m22 m 3 " m " dim r.
To work out the archimedean L-factor for each Langlands parameter ϕ (as in (2.21)) and each irreducible representation of L T (as in (1a),(1b),(2), above), we must compute these representations of W F explicitly enough to be able to recognize them in the classification of irreducible representations given in [Tat79, §2.2.2].

22
The associated representations of W L{F are pµ x˝ϕ qpzq " Following [Tat79, §3.1.1], the L-function of this character of W L{F is Lps, µ x˝ϕ q " Γ R˜s`1 2 Here and below, by pn pmod 2qq we mean the integer 0 or 1 according to the value of n modulo 2. 1b) Suppose x is fixed by G. Then x is as in (2.24), and we have the characters of W L{F Following [Tat79, §3.1.1], the L-function of this character of W L{F is 2) Suppose x is not fixed by G. Then we have x "`a 1 , . . . , a n 1 , c 1 , . . . , c n , pb 1 , b 1 1 q, . . . , pb n 3 , b 1 n 3 q˘, where at least one of the c i ‰ 0 or one of the b i ‰ b 1 i . Then we have In order to find this representation in the classification of [Tat79, §2.2.2], we must recognize it as the induction of some character. We have The power of z in pµ x˝ϕ qpzq iś and the power of z in pµ τ x˝ϕ qpzq is Exactly one of these two is negative. Following Tate, the rule for recognizing which character this representation is induced from is: choose pµ x˝ϕ qpzq or pµ τ x˝ϕ qpzq according to which has a negative power of z. Then the representation is induced off that character. The power of |z| in pµ x˝ϕ qpzq is i and the power of |z| in pµ τ x˝ϕ qpzq is Therefore if the power of z in pµ x˝ϕ qpzq is negative we have that the power of |z| in it is and if the power of z in pµ τ x˝ϕ qpzq is negative we get that the power of |z| in it is given by exactly the same formula. So we find the representation V x˝ϕ of W L{F is induced from the character of W L given by Then by [Tat79, §3.3.1] we can conclude that We now collect the above results in a more compact form. Given a representation r we determine a 3ˆ3 block matrix M " M prq as follows. Take a decomposition where each x i , x 1 i P X are fixed by the action of G and each x 2 i P X is in a G-orbit of cardinality 2. We may write explicitly (2.26) x i " pa i1 , a i2 , . . . , a in 1 , 0, . . . , 0, pb i1 , b i1 q, . . . , pb in 3 , b in 3 qq, and similarly for x 1 i . Likewise we may write (2.27) x 2 i " ppa i1 , a i2 , . . . , a in 1 , c i1 , . . . , c in 2 , pb i1 , b 1 i1 q, . . . , pb in 3 , b 1 in 3 qq, with at least one c ij ‰ 0 or b ij ‰ b 1 ij . Now define the matrix where: A 1 P M m 1ˆn1 pZq is given by A 1 " ppa ij qq with a ij as in (2.26), B 1 P M m 1ˆn3 pZq is given by B 1 " ppb ij qq where b ij is also as in (2.26). Next, A 2 and B 2 are defined similarly to A 1 and B 1 but using the coordinates for x 1 i instead of those of x i as in (2.26). Finally, A 3 P M m 3ˆn1 pZq is given by pa ij q where a ij are taken from (2.27), C P M m 3ˆn2 pZq is given by C " pc ij q where c ij are taken from (2.27), and B 3 P M m 3ˆn3 pZˆZq is given by B 3 " ppb ij , b 1 ij qq, where pb ij , b 1 ij q is also taken from (2.27).
We define block-matrix multiplication as follows. Matrices of the form A multiplied on an element pw, ǫq P piRq n 1ˆp Z{2q n 1 are defined to be where the products on the right hand sides are the usual matrix multiplication. A matrix of the form C multiplied on an element pαq P Z n 2 is defined to by the standard matrix multiplication Cα. We define also for elements pw 1 , α 1 q P C n 3ˆZ n 3 the multiplication B 1 pw 1 , α 1 q " B 1 pw 1´α1 q where on the right hand side we have usual matrix multiplication. We also define the multiplication of B 2 in exactly the same way. So, in summary, Finally, we define the multiplication of pA 3 |C|B 3 q on ppw, ǫq, α, pw 1 , α 1 qq as follows. Let B3 P M m 3ˆn3 pZq be the matrix ppb ij`b 1 ij qq formed from the entries of B 3 and B3 P M m 3ˆn3 pZq the matrix with entries ppb ij´b 1 ij qq. Then we define (2.31) pA 3 |C|B 3 q¨p w, ǫq α pw 1 , α 1 q‚ " A 3 w`B3 pw 1´α1 q`ˇˇCα`B3 pα 1 qˇˇP piRˆNq m 3 , 25 where the absolute values means take the absolute value of each entry. This defines a multiplication If M " M prq is as above and ϕ is a Langlands parameter (2.21), we have where pM ϕq i means the ith entry of M ϕ P piRˆt0, 1uq m 1`m2ˆp iRˆNq m 3 . Let x P T pF q, which according to our chosen isomorphism (2.20) we can express as x Þ Ñ p. . . , x j , . . . , x 1 j , . . . , x 2 j , . . .q P pRˆq n 1ˆp S 1 q n 2ˆp Cˆq n 3 , where x j P Rˆfor j " 1, . . . , n 1 , x 1 j P S 1 for j " 1, . . . , n 2 , and x 2 j P Cˆfor j " 1, . . . , n 3 . Then if χ P T pF q^corresponds to the Langlands parameter ϕ with parameterization (2.21) across the Langlands correspondence (2.4), we have that χ is given explicitly by

Global preliminaries
3.1. Local to global. The main goal of this subsection is the following proposition, which reduces the global counting problem to a local one. Recall for each place v of k the extension K w of k v and corresponding valuation w of K. Let O w and O v be the rings of integers in K w and k v , respectively, and N T pO w q be the image of the norm map N : T pO w q Ñ T pO v q. Let S 8 be the set of archimedean places of k and let be the global norm-units of the torus T . Recall that for all unramified non-archimedean places of k we have N T pO w q " T pO v q by Lemma 2.10, and for all ramified non-archimediean places we have that N T pO w q is a finite-index subgroup of T pO v q by Lemma 2.11. Thus, U N pT q is a finite index subgroup of the global units of the torus U pT q, which by the Dirichlet units theorem for tori [Shy77] is a finitely-generated abelian group.
Proposition 3.1. There exists c P R ą0 , depending only on T and ν so that for any s for which the right hand side converges absolutely.
Proof. We have an exact sequence of locally compact commutative groups where Cl N pT q is a finite group, since the standard class number of T is finite (see [Ono61, Thm. 3.1]) and since ś N T pO w q has finite index in ś T pO v q by Lemma 2.11. By Pontryagin duality and taking the quotient by T pkq, we have a dual exact sequence That is, V pT q is the set of pairs pχ 8 , χ f q, where χ 8 " ś v|8 χ v and χ f " ś v∤8 χ v such that χ 8 pxqχ f pxq " 1 for all x P U N pT q. We givẽ he Haar measure such that each N T pO w q^has counting measure (it is discrete, being the dual of a compact group), and T pk v q^has the Haar measure given by products of counting measure and the standard Lebesgue measure (for more details, see subsection 2.4). Then by [Bou04,VII.44] there is a constant c 1 such that for any integrable function f on ApT q ż Let f pχq " cpχ, rq´s for Repsq sufficiently large. We write each θ " θ 8 θ f " ś v θ v . Since cpχ, rq is defined as a product of local factors, we have for each θ that ż T pk v q^: χ 8 pxq " 1, for all x P U N pT qu, and (3.7) V fin " ź vRS8 N T pO w q^.
Within the integral over V 8 above we mean by χ´1 f any element of ś vPS8 T pk v q^taking the same values as χ f pxq´1 on all x P U N pT q. We have an exact sequence 1 / / U N pT q / / ś vPS8 T pk v q / / V8 / / 1 and by Pontryagin duality 1 / / V 8 / / ś vPS8 T pk v q^/ / U N pT q^/ / 1 . Since U N pT q is discrete, we have that U N pT q^is compact. Therefore, Lemma 3.2 applies with G " ś vPS8 T pk v q^and H " V 8 , and we derive from (3.5) the formula claimed in the statement of the proposition.
In light of Proposition 3.1, there are three distinct cases to consider: (1) Unramified places. In this case, N T pO w q^" T pO v q^by Lemma 2.10, and cpχ v θ v , rq " cpχ v θ v , rq "cpχ v , rq by Theorem 2.8 and Corollary 2.13. Therefore we have (2) Ramified places. In this case we do not have any of the advantages listed in the above unramified case. However, there are only finitely many ramified v, so we need less. We will show in section 5 that converges absolutely in some right half plane with s " A in its interior, where A is the invariant defined in (1.4). We also treat as ramified places any finite place v for which pλ, q kv q ‰ 1, where q kv is the cardinality of the residue field of non-archimedean local field associated to v, and λ is the natural number introduced in the introduction. We call the set of places for which either v is ramified or pλ, q kv q ‰ 1 the "bad" set of places, and denote that set by B.
(3) Archimedean places. If v is archimedean then we have (3.10) A v ps, xq " An expression for this integral via the local Langlands correspondence can be derived from Definition 2.17, and equations (2.33) and (2.34). In terms of the notation just introduced in (3.8),(3.9), and (3.10), the result of Proposition 3.1 reads

3.2.
A global finiteness lemma. Now that the role that the global norm-units U N pT q play in the proof of Theorem 1.1 is apparent, we prove an important global finiteness lemma related to them. As in the introduction, let G " GalpK{kq the global Galois group, and S Ă M be a subset of the co-weights of r. Let G S " Stab G S Ď G. Then G S acts on S, and also on its complement S c . Let K S be the intermediate field in the extension K{k corresponding to G S under the Galois correspondence. Then there is a torus T S defined over K S by taking its cocharacter lattice to be Z |S c | , where the coordinates are indexed by µ P S c , and G S acts by permuting these. We have then a map of group schemes α : T S Ñ T over K S given by the map of cocharacter lattices α˚: Z |S c | Ñ X˚pT q p0, . . . , 1, . . . , 0q Þ Ñ µ, where the 1 is in the µ-slot and the other coordinates are all 0. Now, given a K S -point x of T , we define the (not necessarily irreducible) variety α´1pxq to be the fiber of α over x. Since α´1pxq k is noetherian, it has finitely many irreducible components. Let ppα´1pxqq denote the set of irreducible components. There is a continuous action of G K S on ppα´1pxqq, and therefore there exists a finite extension K 1 x of K S such that all geometric components of α´1pxq are defined over K 1 x . Without loss of generality, we may assume that r K Ď K 1 x . It will be shown later that if x " 1, then all of the components of α´1p1q are in fact defined over r K. On the other hand, the next lemma shows that we can pick a single K 1 {k over which all geometric components of α´1pxq for all x, S are defined.
Lemma 3.3. Let α : T 1 Ñ T 2 be a map of tori defined over a number field k. Every component of α´1pxq for all x P U N pT 2 q is defined over a single finite extension of k. The number of geometric components of α´1pxq is uniformly bounded as a function of x.
Proof. Given α´1pxq, α´1pyq two fibers with fields of definition of their irreducible components K 1 and K 2 , we show that all of the components of α´1pxyq are defined over the compositum K 1 .K 2 .
Suppose more generally that X 1 and X 2 are varieties over k which have all of their irreducible components defined over K 1 and K 2 , respectively. Then, all of the components of X 1ˆX2 are defined over K 1 .K 2 . Indeed, if C is a irreducible component of X 1ˆX2 , then the projection maps π 1 and π 2 send π 1 : C Ñ C 1 and π 2 : C Ñ C 2 , where C 1 and C 2 are irreducible components of X 1 and X 2 . By the universal property, C Ñ C 1ˆC2 , and C 1ˆC2 is defined over K 1 .K 2 , so C is as well.
We have an isomorphism α´1pxqˆα´1pyq » α´1pxyqˆα´1py´1q pu, vq Þ Ñ puv, v´1q, which thus induces a bijection on components. Let K 12 be a field of definition of all the irreducible components of α´1pxyq. Then we have that any component is defined over K 1 .K 2 and also over K 12 .K 2 , so K 1 .K 2 is an extension of K 12 , and thus all components of α´1pxyq are defined over The norm-units U N pT q form a finitely generated abelian group by [Shy77], Lemmas 2.10 and 2.11. Let ζ, ǫ 1 , . . . , ǫ s be generators, and let K 0 , . . . , K s be fields over which all irreducible components of α´1pζq, α´1pǫ 1 q . . . , α´1pǫ s q are defined, respectively. Then we have that all components of all α´1pxq, as x varies over U N pT q are defined over K 0 .¨¨¨.K s , a finite extension of k.
By Lemma 3.3 we may take r K Ď K 1 to be a finite Galois extension of K over which all irreducible components of all α´1pxq for all S, x are defined. Let Γ " GalpK 1 {kq. The finite group Γ acts on the set of irreducible components ppα´1pxqq for any x. We will show later that if x " 1, the action of Γ on α´1p1q factors through Γ Ñ r G. Now suppose p is an unramified prime of k. The valuation w extending v from the first paragraph of this section determines a unique prime P of K lying over p. We also choose for each P, a prime P 1 of K 1 lying over P. Under the restriction map Γ Ñ G, Fr P 1 maps to Fr P .
If Fr P P G fixes S, i.e. Fr P P G S , then T S is defined over k p , the completion of k at p. There exists a model for T over the ring of integers O k of k called the canonical model [Vos98, Ch. IV, §11.2], which we denote by T. The O k -scheme T has the property that for each prime p of O k and each completion k p of k with ring of integers O p we have that TpO p q is the unique (see [Ono61, §2.1]) maximal compact subgroup of T pk p q. Therefore, each of T, T S , and α´1pxq is a variety defined over O p for each prime p of k for which Fr P fixes S, and so each fiber α´1pxq is also defined over the residue field f p of k p and has all of its geometric components defined over the residue field of K 1 P 1 . 3.3. Proof of second assertion of Theorem 1.1.
Theorem 3.4. If r| p T is not faithful, then νptχ P ApT q : cpχ, rq ď Xuq " 8 for some finite X.

29
Proof. Recall the exact sequence (3.3), and in particular the cokernel V pT q " tpχ 8 , χ f q : χ 8 pxqχ f pxq " 1 for all x P U N pT qu, where χ 8 P ś vPS8 T pk v q^and χ f P V fin " ś vRS8 N T pO w q^. For prove Theorem 3.4 it suffices to construct a subset of V pT q of infinite Haar measure on which the analytic conductor remains bounded.
By hypothesis we have that t1u Ĺ ker r| p T . Let S 0 denote the set of unramified places of k which split completely in K{k, and for which q Kw " 1 pmod |π 0 pker r| p T q|q. By the Chebotarev density theorem, |S 0 | " 8. For any v P S 0 we have by Lemma 2.10 and Proposition 2.12 that Write ℓ for the residue field of K w . By construction of S 0 , we have | Hom G pℓˆ, ker r| p T q| ě 2 for any v P S 0 , and therefore | Hom G pOŵ , ker r| p T q| ě 2 as well. All ξ P Hom G pOŵ, ker r| p T q Ď Hom G pOŵ , p T q have r˝pξˆσq of trivial Artin conductor. For any v P S 0 let T pO v q0 be the subset of T pO v qĉ orresponding to Hom G pOŵ , ker r| p T q across the restricted Langlands perfect pairing (2.11). Let V fin,0 Ď V fin be given by t1u.
Then V fin,0 is an infinite set such that every To construct a subset of V pT q of infinite measure on which cpχ, rq is bounded, it suffices to extend each χ f P V fin,0 to V pT q. Recall the notation T pk v q^: χ 8 pxqχ f pxq " 1 for all x P U N pT qu.
Lemma 3.5. There exist constants K, ε ą 0 and a subset Xpχ f q Ď χ´1 f V 8 for each χ f P V fin such that νpXpχ f qq ą ε and supt ź vPS8 c v pχ v , rq : χ 8 P Xpχ f qu ď K for all χ f P V fin .
Proof. We give an explicit description of the sets χ´1 f V 8 in terms of the corresponding Langlands parameters across (2.4). Let x 1 , . . . , x s be generators for U N pT q. Recall T from (2.20). For v P S 8 let σ v : U N pT q ãÑ T pk v q » T be the corresponding embedding. As in subsection 2.4, we write σ v x i " p. . . , x vij , . . . , x 1 vij , . . . , x 2 vij , . . .q, where x vij P Rˆ, x 1 vij P S 1 , and x 2 vij P Cˆ. Consider the image of χ´1 f V 8 across the map where T^was given in (2.22). We write elements of ś vPS8 T^as ppw v , ǫ v q, α v , pw 1 v , α 1 v qq vPS8 . Then the image of χ´1 f V 8 across (3.12) is an affine hyperplane in ś vPS8 T^cut out by for all generators x i , i " 1, . . . , s of U N pT q. Since χ f px i q P S 1 for all χ f and x i , the affine hyperplane in ś vPS8 T^described by (3.13) intersects a fixed (independent of χ f ) compact set around the origin, say U 0 , in a set of positive measure bounded below independently of χ f . For each v P S 8 , the set tχ P T pk v q^: c v pχ, rq ď Xu is in bijection under the local Langlands correspondence with (3.14) H v " tϕ : by the results of section 2.4. Let By the am-gm inequality, we have L v pXq Ď H v pXq. If X is sufficiently large, then L v pXq contains any fixed compact set in T^, and in particular U 0 Ď ś vPS8 L v pXq. Taking K " X |S8| , the lemma is proved.
The fibered set V fin,0ˆX pχ f q Ď V pT q is our candidate for a set of infinite measure and bounded analytic conductor. By Lemma 3.5 and additivity of measure we have νpV fin,0ˆX pχ f qq " ν¨ď yet cpχ, rq ď K for any χ " pχ 8 , χ f q P V fin,0ˆX pχ f q.

Unramified computations
In this section T is a torus defined over a non-archimedean local field F , splitting over an unramified extension L{F with group G " GalpL{F q. Recall from Proposition 2.12 the isomorphism By Proposition 2.9 we have Recall from the introduction the set M of co-weights of r. Let N denote the set of non-negative integers, and let us index the coordinates of N M by µ P M . For each c " pc µ q µPM P N M , consider the following sets of Langlands parameters (restricted to OL ): (4.2) P ď pcq " tξ P Hom G pOL , p T q : cpµ˝ξq ď c µ , for all µ P M u and (4.3) P " pcq " tξ P Hom G pOL , p T q : cpµ˝ξq " c µ , for all µ P M u.

31
The main goal of this section is to give expressions for the coefficients Π " pc, xq because they are the coefficients of U v ps, xq: where |c| " ř µ c µ . We begin our analysis with the sums Π ď pc, xq, since P ď pcq is a finite abelian group. The two functions Π ď pc, xq and Π " pc, xq are related by inclusion-exclusion: Recall that the set of co-weights M admits an action of G. We also let G act on N M by permuting coordinates and let D k pcq be the complex diagonalizable group defined by Note if k " 0 then we recover the groups DpSq from the introduction. If c is G-fixed then D k pcq admits an action of G. Let P be the maximal ideal of L, ℓ the residue field of L, and charpℓq its characteristic.
Proof. Suppose that χ ξ pxq " 1 for all ξ P P ď pcq. Then Π ď pc, xq " |P ď pcq| and it suffices to count the latter set. Since r| p T is faithful, then D k pcq " t1u for sufficiently large k P N, and so there exists k 0 " k 0 pcq " mintk P N : D k pcq " t1uu.
Thus the product in the statement of the proposition is finite, running up to k 0´1 . A parameter ξ is in P ď pcq if and only if ξp1`P k q Ď D k pcq for all k P N. In particular, every ξ P P ď pcq is trivial on 1`P k 0 . We inductively construct all of the ξ P P ď pcq by extending the trivial homomorphism 1`P k 0 Ñ p T backwards along the standard filtration.
Consider two base cases: k 0 " 0 and k 0 " 1. If c is such that k 0 " 0 then D k pcq " t1u for all k P N and P ď pcq " t1u, so the formula in the statement of the proposition holds. If c is such that k 0 " 1 then ξp1`Pq " t1u for all ξ P P ď pcq, and the possible extensions of ξ to OL are parametrized by Hom G pOL {p1`Pq, D 0 pcqq " Hom G pℓˆ, D 0 pcqq.
So the formula in the statement of the proposition holds. Now suppose as the induction hypothesis that for all c such that k 0 ď K. Consider c such that k 0 " K`1. Then all ξ P P ď pcq satisfy ξp1`P K`1 q " t1u, and the possible extensions the trivial map 1`P K`1 Ñ p T to elements of Hom G p1`P K , D K pcqq are parameterized by since L{F is unramified. Therefore (4.8) holds for c such that k 0 " K`1. By induction, (4.8) holds for all c P N M .
By the normal basis theorem, there exists α P ℓ such that tα, α q F , α q 2 F , . . . , α q rL:F s´1 If there exists ξ P P ď pcq such that χ ξ pxq ‰ 1, then it immediately follows from orthogonality of characters that Π ď pc, xq " 0, hence the second part of the proposition.
Proposition 4.1 is only valid for G-fixed c P N M (since otherwise D k pcq is not a G-module, and G-equivariant homomorphisms into D k pcq do not make any sense). However, we can always reduce to the case that c is G-fixed by the following Lemma.
Later, in Lemma 7.4, we shall see that only the c P N M all of whose entries are 0 or 1 will matter for the location and order of the rightmost pole of Zpsq. Therefore we spend the rest of the section devoting particular attention to this case.
If all of the entries of c are 0 or 1, then D k pcq " t1u for all k ě 1. Therefore we restrict our attention to the case k " 0. We make a change of variables, and instead consider subsets S Ď M as in the introduction. The change of variables is given by the G-equivariant bijection t0, 1u m » 2 M (4.9) c Ø tµ : c µ " 1u, (4.10) with G acting on 2 M as in the introduction. Define the quantities Π ď pS, xq and Π " pS, xq via the above bijection c Ø S in terms of Π ď pc, xq and Π " pc, xq, and define DpSq " D 0 pcq as in the introduction.
Let Fr P G denote the Frobenius element. By Lemma 4.2, it is no loss of generality to suppose that Fr S " S. Recall from section 3 the map of tori α : T S Ñ T . We saw in that section that the fiber α´1pxq is a variety defined over the residue field f of F , and that there is a finite Galois extension L 1 of L such that all geometric components of α´1pxq for all x P U N pT q are defined over the residue field ℓ 1 of L 1 . Let ppα´1pxqq be the finite set of geometric components. There is a continuous action of GalpL 1 {F q on ppα´1pxqq. Let Fr 1 P GalpL 1 {F q denote a Frobenius automorphism, and write (4.11) apS, xq " #ty P ppα´1pxqq : Fr 1 y " yu.
The number apS, xq does not depend on the choice of Fr 1 , since the inertia subgroup of GalpL 1 {F q acts trivially on α´1pxq.
Lemma 4.3. Suppose Fr S " S, and that L{F is unramifed. Then bound", 31 Aug, 2012. Define the "complexity" of a variety V as follows. Let f denote an algebraic closure of a finite field f . If V is defined over f by V " tx P f d : P 1 pxq " . . . " P m pxq " 0u, then the complexity C is defined to be the maximum of d, m, and the degrees of the P i .
Theorem 4.4 (Lang-Weil). Let V be a variety of complexity at most C defined over f . Then one has |V pf q| " pppV q`O C p|f |´1 {2 qq|f | dimpV q where ppV q is the number of geometrically irreducible components of V of dimension dimpV q that are invariant with respect to the Frobenius endomorphism x Þ Ñ x |f | associated to f .
If in fact the dimension of V is zero, it is not hard to see that |V pf q| " ppV q. We will apply the Lang-Weil theorem to the varieties α´1pxq, as x varies over global units of T . We have that the complexity of α´1pxq is bounded uniformly as x varies, in terms of the degree of the equations cutting out T S and α, and dim T and dim r. Since there are only finitely many possibilities for S for a given T, r, the complexity depends only on T, r, and so the error term also depends only on T, r.
By e.g. [Mil17, Thm. 1.72, Def. 1.73], the map α factors as α : T S Ñ αpT S q Ñ T , with the first map faithfully flat and the second a closed immersion. For any point x P αpT S q, we have by e.g. [Mil17,A.73] that dim α´1pxq`dim αpT S q " dim T S , and by [Mil17,Rem. 5.42] that dim DpSq " dim T´dim αpT S q.
From the Lang-Weil theorem, we conclude the lemma.
In the special case that the global unit x " 1 we state the leading constant in Lemma 4.3 in a more convenient fashion. Let Fr P G and (4.12) apSq " |ty P π 0 pDpSqq : Fr y q F " yu|.
Proof. We give an alternate computation of | Hom G pℓˆ, p T q|. Let x denote a generator for the cyclic group ℓˆ. Then Hom G pℓˆ, DpSqq is in bijection with the set tz P DpSq : Fr z " z q F u of possible images of x in DpSq. This set is equal to the kernel K of the G-equivariant homomorphism We have an exact sequence of G-modules The map ϕ is given by ϕpχq " q F χ´χ Fr , where we have written X˚pDpSqq in additive notation. Our goal is to compute the cardinality of X˚pKq, which equals the cardinality of K itself. Write X " X˚pDpSqq, X t for the torsion subgroup, and X f " X{X t . The map ϕ : X Ñ X induces maps X t Ñ X t and X f Ñ X f , both of which we also denote ϕ. We write Q " X˚pKq The lower right horizontal arrow is surjective since both arrows on the other side of the square it forms are both surjective.
We show that the top right ϕ is injective. Indeed, let χ P X f satisfy ϕpχq " 0. Since ϕ is G-equivariant, we also have ϕpχ Fr i q " 0, for all i. Since ϕpχq " 0 we have q F χ " χ Fr , and so χ Fr " 0 pmod q F q. But similarly, since ϕpχ Fr q " 0 we have that χ Fr 2 " 0 pmod q 2 F q. Therefore χ " 0 pmod q |G| F q. Repeating this process ad infinitum, we conclude that χ " 0 P X f , so the top right ϕ is injective.
Then by the snake lemma we have that Q t ãÑ Q, and so the bottom row of the diagram forms an exact sequence of finitely-generated abelian groups. We have that Q is finite if both Q t and Q f are, and in this case |Q| " |Q t ||Q f |.
Let us begin with Q f . The map ϕ on X f is given in matrices by q F I´A, where A is some matrix of integers for which A |G| " I. Putting q F I´A in Smith normal form q F I´A " U DV with U, V P GL dim DpSq pZq, we have with each d i is equal to q F˘1 , and so |Q f | " q dim DpSq F p1`Opq´1 F qq. Now we compute |Q t |. For any endomorphism of a finite abelian group f : A Ñ A, we have that | ker f | " | coker f |. Let K t be the kernel of ϕ : X t Ñ X t , which therefore has the same cardinality as Q t . But the cardinality of K t is exactly the quantity apSq defined above the statement of the Lemma.
We have shown that |Q t | " apSq and |Q f | " q dim DpSq F p1`Opq´1 F qq, and since |Q| " |Q t ||Q f |, we conclude the lemma.
Finally, we apply the foregoing results on Π ď pS, xq to derive the final results for Π " pS, xq. Let The main tool is (4.6), which we re-state for the sets S as (4.13) Π " pS, xq " ÿ T ĎS µp1 P T q¨¨¨µpm P T qΠ ď pS´T, xq.
For a set S we denote (4.14) S red " tµ P S : σµ P S for all σ P Gu.
Recall from (1.4) the positive rational A.
Lemma 4.6. For any ∅ ‰ S Ď M such that Fr S " S, and Proof. Suppose first that dim DpSq " 0 and apS, xq ‰ 0. Then for any T Ď S we also have dim DpT q " 0 since DpT q Ď DpSq. By (4.13) and (4.15) we have µp1 P T q¨¨¨µpm P T qΠ ď ppS´T q red , xq.
Since apS, xq ‰ 0, we have by Lemma 4.3 and Proposition 4.1 that χ ξ pxq " 1 for all ξ P P ď pSq. Then for any S 1 Ď S we also have χ ξ pxq " 1 for all ξ P P ď pS 1 q. Using Proposition 4.1 and Lemma 4.3 again, we have Π " pS, xq " apS, xq`ÿ ∅‰T ĎS µp1 P T q¨¨¨µpm P T qappS´T q red , xq.

For any
thus DpT q " t1u by definition of A. For all T ‰ ∅, we have pS´T q red Ĺ S. Thus, if pS´T q red ‰ ∅, then we have DppS´T q red q " t1u. On the other hand, if pS´T q red " ∅, then we also have DppS´T q red q " t1u by the fidelity of r| p T . Therefore Π " pS, xq " apS, xq`ÿ ∅‰T ĎS µp1 P T q¨¨¨µpm P T q " apS, xq´1`ÿ T ĎS µp1 P T q¨¨¨µpm P T q " apS, xq´1, since we assumed S ‰ ∅. Now suppose that dim DpSq ě 1 and apS, xq ‰ 0. As above, we have by (4.13), (4.15), Proposition 4.1 and Lemma 4.3 that as q F Ñ 8. Suppose that T is such that dim DppS´T q red q " dim DpSq. If pS´T q red " ∅, then 1 ď dim DpSq " dim DppS´T q red q " 0, which is a contradiction with the fidelity of r| p T . Therefore we may assume that pS´T q red ‰ ∅. If T ‰ ∅ then dim DppS´T q red q`1 |pS´T q red | " dim DpSq`1 |pS´T q red | ą dim DpSq`1 |S| ě A.
Therefore DppS´T q red q " t1u, and this is a contradiction with dim DppS´T q red q " dim DpSq. Thus, the only T Ď S which satisfies dim DppS´T q red q " dim DpSq is T " ∅, from which we conclude the statement in the lemma. Now suppose apS, xq " 0. If dim DpSq ě 1 then by (4.13), Lemma 4.3 and the triangle inequality, the statement of the lemma holds.
To finish the proof of lemma, it remains to consider the case that dim DpSq " 0. If DpSq " t1u, then we must have apS, xq ‰ 0, so suppose that dim DpSq " 0 and DpSq ‰ t1u. Suppose |S| ě 2 and S is maximal such that dim DpSq " 0 and DpSq ‰ t1u. There exists µ R S, since DpM q " p T . We claim that dim DpS Y µq ě 1. Indeed, by maximality, either dim DpS Y µq ě 1 or DpS Y µq " t1u. But the second of these can't happen since DpSq ‰ t1u already, and Dp¨q is monotonic. So dim DpS Y µq ě 1. But then since |S| ě 2. This is a contradiction with the definition of A. Hence, |S| " 1. Then we have Here, Π ď pS, xq " 0 by the assumption that there exists ξ P P ď pSq with χ ξ pxq ‰ 1, and Π ď p∅, xq " 1 since only the trivial character appears in the definition of Π ď p∅, xq.

Ramified computations
In this section, T is a torus defined over a non-archimedean local field F and splitting over a finite Galois extension L. Let G " GalpL{F q be the corresponding group. Let p, P be the maximal ideals of F, L, respectively.
By Theorem 2.8, Corollary 2.13, and Proposition 2.9 we have The conductor c appearing in the last line of (5.1) is in fact c " c U as in Definition 2.3, where U is the standard filtration on OL (see Definition 2.5). Next, we construct yet another filtration and compare it to U.
By the normal basis theorem, there exists an element α P L such that tα g : g P Gu is a basis for L{F . The tα g u all have the same valuation (e.g. [Ser79, Ch.2 Cor 3]), so by clearing numerators or denominators, there exists β P OL such that tβ g : g P Gu is a basis for L. We define an injective map of O F rGs-modules f : O F rGs ãÑ O L by f p1q " β. Its image is finite-index in O L since tβ g u span L.
Let ν ě 1 be sufficiently large so that the p-adic exponential function exp : P ν Ñ 1`P ν is well-defined and an isomorphism. Then let g : O F rGs ãÑ OL be defined as the composition of the following sequence of injective maps The homomorphism g has finite cokernel. Let V n " gpp n O F rGsq and V " pV n q be the corresponding filtration of OL . We have for all n ě 0 that where e L{F is the ramification index of L{F and ν is as above. Indeed, if x P p n O F rGs then we write x as x " ÿ gPG a g g with a g P p n for all g P G. So we have f pxq " ÿ gPG a g β g P p n " P e L{F n , so that f pp n O F rGsq Ď P e L{F n . Now let us consider the conductor c V defined with respect to the filtration V and compare c U and c V . Let χ : OL Ñ Cˆbe a character. If χ| O pnq L " 1 then c U pχq ď n, and if χ| O pnq L ‰ 1 then c U pχq ě n`1. Similarly, if χ| V n " 1 then c V pχq ď n, and if χ| V n ‰ 1 then c V pχq ě n`1. Therefore by (5.2) we have c U pχq ě e L{F c V pχq`ν´e L{F`1 .
Then we have ÿ But now the summand only depends on the restriction of ξ to V 0 . We have an exact sequence he kernel is a finite group, since OL {V 0 is a finite group and p T only has finitely many points of order dividing the cardinality of this group. Thus we have ÿ

But also
Hom If ξ Ø τ P HompO F , p T q across this isomorphism, then c V pµ˝ξq " c W pµ˝τ q, where the latter is the conductor with respect to the filtration W " pp n q of the additive group O F . Therefore Then one computes similarly to section 4. Let Π ď pcq " |tτ P HompO F , p T q : c W pµ˝τ q ď c µ for all µ P M u|.
If r| p T is faithful and c P N M is G-fixed then By the fidelity of r| p T , the product is actually a finite product. We have therefore that Therefore, R v ps, xq converges absolutely and uniformly on compacts for all

Archimedean computations
Recall (3.10) for v an archimedean place of k, x P T pk v q that Assume that k v » R, and recall (2.20) that we have chosen an isomorphism (6.1) T pk v q » T " pRˆq n 1ˆp S 1 q n 2ˆp Cˆq n 3 x Þ Ñ p. . . , x j , . . . , x 1 j , . . . , x 2 j , . . .q, with x j P Rˆ, x 1 j P S 1 , and x 2 j P Cˆ. The case that k v » C is very similar to the situation that n 1 " n 2 " 0 above, so we ignore it and assume that k v » R for the remainder of this section.
(2) For s in the above region of absolute convergence, we have with at most polynomial growth in s in a vertical strip.
The proof of Theorem 6.1 will occupy the remainder of section 6 of this paper. In subsection 6.1 we reduce assertion (1) to a problem in combinatorial geometry (see Proposition 6.3). The main input in the proof of assertion (1) is a Brascamp-Lieb inequality, which is a heavy tool from pure analysis. In subsection 6.2, we give some background information on matroids and polymatroids, and in subsection 6.3 we solve the combinatorial geometry problem. Assertion (2) follows immediately. Finally, in subsection 6.4 we prove assertion (3) of Theorem 6.1.
Recall that in subsection 2.4 we worked out explicitly the local Langlands correspondence for tori over archimedean local fields. Specifically, in (2.21) we explicitly parameterized (with respect to choices K w » C and (6.1)) equivalence classes of Langlands parameters ϕ : W L{F Ñ L T u by ppw, ǫq, α, pw 1 , α 1 qq P piR n 1ˆp Z{2Zq n 1 qˆZ n 2ˆp iR n 3ˆZ n 3 q " T^.
6.1. Convergence. We apply the triangle inequality A v ps, xq. For i " 1, . . . , m 1`m2 , the pM ϕq i P iR or iR`1 by inspection of (2.29), (2.30). For such i we apply the inequality 1 ?
Then we make the change of variables w j Þ Ñ iw j and w 1 j´α 1 j Þ Ñ iw 1 j , so that χ P T pk v q^unitary implies that w j , w 1 j P R. We introduce some notation to record the result (see (6.3)) of the aforementioned manipulations of A v ps, xq. Let M re denote the pm 1`m2`m3 qˆpn 1`n3 q matrix were A 1 , A 2 , A 3 , B 1 , B 2 , B3 were defined in subsection 2.4. Such a matrix acts on w " pw, w 1 q P R n 1`n3 by the usual multiplication of matrices. Let also where C and B3 were also defined in subsection 2.4. The integral m 3ˆp n 2`n3 q matrix M int acts on α " pα, α 1 q P Z n 2`n3 by the usual multiplication of matrices. The result of our inequalities and changes of variable is or, rearranging, Before proceeding with the estimation of (6.3) or (6.4), we first describe a result in combinatorial geometry. Let M P M mˆn pRq be an mˆn matrix with real entries.
Definition 6.2. For any α ě β ě 1 we say that M is pα, βq-biased if there exist α rows of M such that any basis of R n formed from rows of M contains at least β of the distinguished α rows.
For example, note that any full-rank mˆn matrix is pm, nq-biased. Let us now write a i , i " 1, . . . , m for the rows of M , and consider the convex polytope H M cut out by the the following inequalities on R m ě0 : (6.5) m ÿ i"1 x i " n and (6.6) ÿ iPS x i ď dimpspanpta i : i P Suqq for every subset S Ď t1, . . . , mu. Write }¨} 8 for the L 8 -norm on R m , i.e. for x P R m we set The norm }¨} 8 is a convex and piecewise-linear function on R m . Let (6.7) Proposition 6.3. Let M be any full-rank mˆn matrix with real entries. We have that The proof of Proposition 6.3 will be given in subsection 6.3. We now give the proof of assertion (1) of Theorem 6.1, assuming Proposition 6.3.
We estimate A v ps, xq starting with the integral over w in the interior of (6.3). The main tool to bound such an integral is a Brascamp-Lieb inequality. The following necessary and sufficient conditions were first proven in [Bar98], and then re-stated in the form below by [CLL04,§4].
Theorem 6.4 (Brascamp-Lieb Inequality). Let a 1 , . . . , a m be non-zero vectors in R n which span R n , and let M be the mˆn matrix whose rows are a i . Let p " pp´1 1 , . . . , p´1 m q P R m ě0 . Let f " pf i q i"1,...,m be an m-tuple of non-negative measurable functions f i : R Ñ R ě0 . Then ż Here the implied constant depends on m, n, M, p, but not on f . Note that H Mre is compact, so the infimum in (6.7) is attained, say by B " pB 1 , . . . , B m q P H Mre . Note that r| p T faithful implies that M re is full-rank. We apply Theorem 6.4 to the interior integral over w of (6.3) with M " M re , p " B, Now we need to bound the sum over α in (6.8). Luckily, a version of the Brascamp-Lieb inequality on Z n has been given by [BCCT10,Thm. 2.4].

42
Theorem 6.5 (Bennett, Carbery, Christ and Tao). Let G and tG i : 1 ď i ď mu be finitely generated Abelian groups. Let ϕ i : G Ñ G i be homomorphisms. Let p i P r1, 8s. Then x i ď rankpspan Z pta i : i P Suqq for every subset S Ď t1, . . . , mu, by tensoring with R. The discussion on [BCCT10,p. 649] shows that (6.11) and (6.12) imply that where ϕ i : Z n 2`n3 Ñ Z is given by x Þ Ñ xa i , xy. The infimum in (6.7) is attained, say by B 1 " pB 1 1 , . . . , B 1 m q P H M int . We apply Theorem 6.5 with G " Z n 2`n3 , G i " Z, ϕ i as above, p i " B 1 i´1 , and f i pxq " } 1 p ?
By Proposition 6.3 (or, a minor variation thereof), we have that A v ps, xq converges absolutely when Repsq ą maxpmaxt β α 1`2 α 2 : M re is pα, βq-biasedu, 1 2 maxt β α : M int is pα, βq-biaseduq, 43 where in the first max α " α 1`α2 , and the distinguished set (in the definition of pα, βq-bias) contains α 1 of the first m 1`m2 rows and α 2 of the last m 3 rows. Let We claim that (6.18) maxpmaxt β α 1`2 α 2 : M re is pα, βq-biasedu, Indeed, suppose the first maximum on the left hand side is larger. Then there are α " α 1`α2 distinguished rows of M re , α 1 of which are among the first m 1`m2 rows, and α 2 are among the last m 3 rows. Choose the corresponding α 1 rows of M 1 among the first m 1`m2 rows, and the corresponding 2α 2 rows, i.e. α 2 pairs of rows of M 1 from among the last 2m 3 rows. This set of α 1`2 α 2 rows of M 1 shows that M 1 is pα 1`2 α 2 , βq-biased. So Similarly, suppose the second maximum on the left hand side of (6.18) is larger. Then there are α distinguished rows of M int , and we choose the corresponding 2α rows, i.e. α pairs of rows from among the last 2m 3 rows of M 1 . This distinguished set of 2α rows of M 1 shows that M 1 is p2α, βq-biased. So which finishes the proof of (6.18). Finally, there is a bijection between the rows of M 1 and the coweights of r via the isomorphism (2.23). Under this isomorphism, sets of rows of M 1 correspond bijectively to subsets S Ď M of coweights of r, and |S| " α and dim DpSq " β. This concludes the proof of assertion (1) of Theorem 6.1.
Assertion (2) of Theorem 6.1 follows immediately. Indeed, returning to equation (6.2), we integrate by parts once in each variable w i , w 1 i , and apply part (1) of the theorem.
6.2. Background on matroids and polymatroids. The key observation in the proof of Proposition 6.3 is that the definition of pα, βq-bias makes sense more generally for matroids, and H M is exactly the matroid base polytope associated to the matroid M . To this end, we next recall some background on matroids and polymatroids. The following exposition was communicated to the author by R. Zenklusen.
Definition 6.6 (Matroid). A matroid is a pair pN, Iq where N is a finite set and I Ă 2 N is a family of "independent" subsets of N satisfying the following axioms.
(1) I ‰ ∅ (2) If I P I and J Ă I then J P I.
(3) If I, J P I and |J| ą |I| then there exists e P JzI such that I Y teu P I.
Example 6.7 (Linear Matroid). If N is a set of vectors spanning a vector space, and I is the set of linearly independent subsets of N , then pN, Iq is a matroid. One calls such a matroid a linear matroid.
If pN, Iq is a matroid, then the set of bases B Ă I is the set of maximal subsets of I, ordered by inclusion. If pN, Iq is a linear matroid, then B consists of subsets of vectors which form a basis.
Definition 6.8. The rank function of a matroid is the function r : 2 N Ñ Z ě0 given by rpSq " maxt|I| : I P I, I Ă Su.
If pN, Iq is a linear matroid, then rpSq is the dimension of the space spanned by the vectors in S. By the definitions of B and r we have B " tI P I : rpIq " rpN qu.
Let pN, Iq be a matroid and let 1 I P t0, 1u N be the indicator function of I. If S is a finite set of points in R N , then we write convpSq for the convex hull formed from those points.
Definition 6.9. The matroid polytope of pN, Iq is P I " convpt1 I : I P Iuq Ă R N and the matroid base polytope is Lemma 6.10. The rank function of a matroid pN, Iq satisfies the following properties.
‚ r : 2 N Ñ Z ‚ r is submodular: rpAq`rpBq ě rpA Y Bq`rpA X Bq ‚ r is monotone: rpAq ě rpBq for all B Ď A Ď N ‚ r is non-negative: rpAq ě 0 for all A Ď N ‚ r satisfies rpA Y teuq ď rpAq`1 for all A Ď N and e P N . If r is any function enjoying these 5 properties, then there exists a unique matroid whose rank function is r.
We can also express the matroid polytope and matroid base polytope in terms of the rank function as follows. Let x P R N ě0 , e P N and x e be the e-th component of x. For a subset S Ď N we set xpSq " ř ePS x e . In terms of xpSq, we have (see [Sch03,Cor. 40.2b]) P I " tx P R N ě0 : xpSq ď rpSq for all S Ď N u. Then P B is one face of the matroid polytope P I given by a supporting hyperplane, i.e. we have (see [Sch03,Cor. 40.2d]) (6.19) P B " P I X tx P R N : xpN q " rpN qu.
Theorem 6.11 (Matroid Intersection). Let pN, I 1 q and pN, I 2 q be two matroids on the same ground set. Then we have maxt|I| : I P I 1 X I 2 u " min AĎN tr 1 pAq`r 2 pN zAqu.
Definition 6.12 (Polymatroid). A polymatroid on N is a polytope P f " tx P R N ě0 : xpSq ď f pSq for all S Ď N u where f : 2 N Ñ R ě0 is a submodular and monotone function.
The following definition generalizes Definition 6.2 from linear matroids to matroids.
Definition 6.14 (pα, βq-bias for matroids). We say a matroid pN, Iq is pα, βq-biased if there exists S Ď N with |S| " α and such that |B X S| ě β for all bases B Ď N .
Lemma 6.15. A subset S Ď N satisfies rpN q´rpN zSq ě β if and only if for any basis B Ď N we have |B X S| ě β.
Proof. "Only if": Let B Ď N be any basis. By sub-modularity of the rank function we have rpB X Sq`rpN zSq ě rpN q, but B X S is independent, so we have |B X S| " rpB X Sq ě rpN q´rpN zSq ě β.
"If": Suppose that B P B is such that |B X S| is minimal as we range over all bases. Equivalently, B is such that |BzpB X Sq| is maximal. We claim that BzpB X Sq is maximal by inclusion among independent sets which are disjoint from S. From this claim it follows by definition of the rank function that |BzpB X Sq| " rpN zSq, and so rpN q´rpN zSq " |B X S| ě β.
If the claim were false, then there would exist e R S and R BzpB X Sq such that (6.20) pBzpB X Sqq Y teu " pB Y teuqzpB X Sq is an independent set, by matroid axiom (3). Since the set in (6.20) is independent and rpB Yteuq " rpN q, we can complete it to a basis r B Ď B Y teu. But then we have pB Y teuqzpB X Sq " r Bzp r B X Sq, from which it follows that | r Bzp r B X Sq| " |pB Y teuqzpB X Sq| " |BzpB X Sq|`1.
This contradicts the minimality of |B X S| among all bases B P B. Therefore the claim is true.
6.3. Proof of Proposition 6.3. Considering the level sets of the L 8 norm, the B 8 defined in (6.7) becomes B 8 " inftλ ě 0 : P B X r0, λs N ‰ ∅u. (Aside: compare this and (6.19) to the discussion of the Manin conjecture in subsection 1.2.) From the description (6.19) of the matroid basis polytope in terms of the rank function, we have B 8 " inftλ ě 0 : suptxpN q : x P P I X r0, λs N u " rpN qu.
Since this last min is always ď rpN q, to characterize B 8 it suffices to find the smallest λ ě 0 such that for all A Ď N (6.21) rpAq`λ|N zA| ě rpN q, that is to say It changes nothing to swap A with N zA, so B 8 " inftλ ě 0 : rpN zAq`λ|A| ě rpN q for all A Ď N u.
If A " ∅ then the inequality is satisfied for all λ, so suppose not. We then have by Corollary 6.16 6.4. Positivity. To prove assertion (3) of Theorem 6.1 we first establish one-variable versions of the result.
Lemma 6.17. For all real 0 ă σ ď 2 the Fourier transforms of the following functions are positive or`8: f pxq " 1 p ?
Proof. When ξ ą 0 we have by contour shifting The value p f p0q is clearly positive if it converges, and if ξ ă 0 we follow the same steps as above, shifting the contour up instead of down. Now we show p gpξq is positive or`8. For a real parameter 0 ď β ď 1 define (6.22) p g β pξq " x 2`β2`1 q σ dx, so that p gpξq " p g 0 pξq´p g 1 pξq. We have (6.23) p gpξq " x 2`β2 p a x 2`β2`1 q σ`1 dx dβ.
It follows from Lemma 6.17 that the Fourier transforms of 1 p|x|`1q σ , and 1 p|x|`1q σ`1 p ?
The factor inside the parentheses above is positive by the mean value theorem. The Fejér kernel is also positive, and therefore the series is positive wherever it converges. Lemma 6.19. For all real 0 ă σ ď 2 and ξ P Cˆ, the function ÿ βPZˆ| ξ| ξ˙β ż R ep´|ξ|xq p a x 2`β2`1 q σ dx is positive or`8.
Proof. By summation by parts twice, it suffices to show that the second forward difference in β of is positive. Recall the definition of p g β pξq from (6.22), in terms of which we have f pβ`2, ξq´2f pβ`1, ξq`f pβ, ξq " As in the proof of Lemma 6.17, we have x 2`α2 p ?
We have that the Fourier series/transform of each of these is also a tempered distribution: Fpf i q P S 1 pTq. By Lemmas (6.17), (6.18), and (6.19), each of Fpf i q is a positive distribution for σ 0 ă σ ď 2. By assertion (1) of Theorem 6.1 and (6.2), the pm 1`m2`m3 q-fold convolution of the distributions Fpf i q is defined and we have for all x P T. Since the convolution of positive distributions is positive, we have that A v pσ, xq takes positive values for σ 0 ă σ ď 2 and all x P T pk v q, as was to be shown.

Counting
In this section we prove the analytic continuation of the generating series Zpsq. Following Tate [Tat79, §3.5] we take ψ to be a non-trivial additive character of A{k and dx the Haar measure on A such that ş A{k dx " 1. Let ψ v be the local component of ψ at a place v, dx " ś v dx v be any factorization of dx into a product of local measures such that the ring of integers O v at all but finitely many v gets measure 1, and δpψ v q be the function defined in [Tat79, §3.4.5]. Recall the notation of (3.8), (3.9), and (3.10), and set To determine the analytic properties of Zpsq, it suffices to determine the analytic properties of Aps, xq, U ps, xq, and Rps, xq, and to show that sum over x P U N pT q in (7.1) converges absolutely. 7.1. Unramified places.
(1) The series U ps, xq converges absolutely and uniformly on compacta in the right half-plane Repsq ą A, and admits a meromorphic continuation to the right half-plane Repsq ą A´p2m 2 q´1 with at most a pole at s " A.
(2) The series U ps, 1q has a pole at s " A of order | r Gz r Σ 0 | with positive leading constant in its Laurent series expansion.
(3) The possible pole of U ps, xq at s " A is of order ď | r Gz r Σ 0 |, has positive leading coefficient, and Laurent series expansion bounded by that of U ps, 1q at s " A.
(4) Away from a small neighborhood N surrounding s " A, we have the bound U ps, xq ! N,Repsq p1`|s|q K , uniformly in x, for some constant K depending only on T and r.
We devote the rest of this subsection to proving Theorem 7.1. Since our computations in this section are global in nature, we switch to the notation of prime ideals. To each place v R S 8 Y B there is associated a prime ideal p of k, and we have q kv " N ppq, the absolute ideal norm of p. Recall from section 3 the (global) Galois extensions k Ď K Ď K 1 with groups Γ " GalpK 1 {kq and G " GalpK{kq. Recall from section 3 we have chosen for each prime p of k a prime P 1 of K 1 lying over P. Let D P Ď G and D P 1 Ď Γ be decomposition groups at p. The results of section 4 apply with D P and D P 1 playing the role of the local Galois groups.
We have the restriction map Γ Ñ G under which Fr P 1 Þ Ñ Fr P . When x " 1 P U N pT q we will also use the map r G ãÑ GˆGalpkpζ λ q{kq Fr P 1 Þ Ñ pFr P , N ppqq, where N ppq P Galpkpζ λ q{kq denotes the automorphism of kpζ λ q sending a primitive λth root of unity ζ λ to ζ N ppq λ (see (1.8)). In particular, the value of N ppq modulo λ is determined by Fr P 1 P r G. By (4.1) and (4.5) we have U ps, xq " ź pRB ÿ cPN M Π " pc, xq N ppq s|c| .
Definition 7.2. If F 1 psq and F 2 psq are meromorphic functions defined in Repsq ą σ i , i " 1, 2 and there exists an analytic function Gpsq given by an absolutely and uniformly convergent Euler product in Repsq ą σ 0 such that F 1 psq " GpsqF 2 psq, then we say that F 1 equals F 2 up to an absolutely convergent Euler product in Repsq ą σ 0 and write F 1 « F 2 in Repsq ą σ 0 .
Lemma 7.3. Suppose F 1 and F 2 are as in Definition 7.2.
The proofs are easy exercises, so we omit them. Since we do not give an expression for the leading constant in Theorem 1.1, it suffices to study U ps, xq up to the « equivalence.
Proof. It suffices to show that the series (7.2) ź pRB ř cPN M Π " pc, xqN ppq´s |c| ř cPt0,1u M Π " pc, xqN ppq´s |c| converges absolutely and uniformly on compacta in Repsq ą A´1 2m 2 . Consider p R B with N ppq sufficiently large. The factor of the product (7.2) at p is onsider the indices c P N M appearing in the first sum on the right hand side. We may assume that c is such that Π " pc, xq ‰ 0. In particular, it is D P -fixed by Lemma 4.2.
We have the trivial bounds |Π " pc, xq| ď Π " pc, 1q ď Π ď pc, 1q and by Lemmas 4.1 and 4.3 the bound Π ď pc, 1q ! T,r Therefore we have e.g. for the first sum in (7.3) that ÿ cPN M max cµě2 Π " pc, xq N ppq s|c| ! ÿ c : max cµě2 Π"pc,xq‰0 8 ź k"0 N ppq dim D k pcq´s|tµ:cµąku| . By Lemma 7.5 we have the product here has at least two non-one factors for each c in the outer sum. Now we take the product of (7.3) over p R B, and find that (7.2) converges absolutely and uniformly on compacta in the region (7.6) sup iě2 max 1ďjďi S j ĎM t dim DpS 1 q`¨¨¨`dim DpS i q`1 |S 1 |`¨¨¨`|S i | : DpS j q ‰ t1uu.
Lemma 7.6. For any integers a, a 1 and any 1 ď c, c 1 ď m one has a`a 1`1 c`c 1 ď maxt a`1 c , a 1`1 c 1 u´p2m 2 q´1. Proof. Elementary. By Lemma 7.6, we conclude that (7.2) converges absolutely and uniformly in the region Repsq ą A´p2m 2 q´1.
Consider the interior product of (7.7).
Proof. By Lemma 4.2 we have if Fr P S ‰ S then Π " pS, xq " 0. Consider the sets ∅ ‰ S Ď M which satisfy Fr P S " S, and take p2 M zt∅uq D P " Σ P \ Σ P c .
Recall the quantity apS, xq from (4.11). Similarly, let (7.10) a C pS, xq " #ty P ppα´1pxqq : Fr P 1 y " yu, which only depends on the conjugacy class C Ď Γ of Fr P 1 . We have by Lemmas 7.4, 7.7 and 4.6 (7.11) U ps, xq « ź CĎΓ ź pRB n Repsq ą A´p2m 2 q´1. Note that if dim DpSq`1 |S| ą A then DpSq " t1u by the definition of A, and so a C pS, xq " 1 and the term corresponding to S above vanishes.
We split up the sum in (7.11) over the possible values of dim DpSq, |S|. The parameter space is P " tpa, bq : a ě 0, b ě 1, a`1 b " Au.

54
The group Γ acts on each r Σ a,b through G on S and through its Galois action on ppα´1pxqq. If x " 1, the action factors through the action of pg, g˚q P GˆpZ{λZqˆon elements pS, yq with y P π 0 pDpSqq (see Lemma 4.5) is pg, g˚q.y " gy g˚.
Let V a,b be the permutation representation of Γ acting on r Σ a,b . Let ψ a,b be its character, and C a conjugacy class of Γ. Then ψ a,b pCq is the number of Fr P 1 -fixed points on r Σ a,b . In these terms, we have Since Lp1, χ i q ‰ 0 for any Hecke characters χ i , the number of poles at s " A appearing in (7.15) is equal to the number of trivial characters appearing among the Hecke characters χ i , counted with multiplicity. By e.g. Serre [Ser77, ex. 2.6], the number of trivial characters is equal to the number of orbits ÿ a,bPP |Γz r Σ a,b |.
If x " 1 then this matches | r Gz r Σ 0 | as defined in the introduction. Thus we have established part (2) of Theorem 7.1.