Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip

Let $\pi$ and $\pi_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given family and $\pi_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion in class groups of number fields.

The generalized Riemann hypothesis (GRH) for Dirichlet L-functions implies that if a and q ≥ 1 are coprime integers, then there exists a prime 1 p ≪ (q log q) 2 such that p ≡ a (mod q). Linnik [45] unconditionally proved that the least such prime is O(q A ), where A > 0 is an absolute and effective constant; up to the quality of A, Linnik's result is commensurate with what GRH predicts. Linnik's proof developed powerful results for the distribution of zeros of Dirichlet L-functions near the point s = 1, including a log-free zero density estimate. In this paper, we prove a flexible log-free zero density estimate for families of L-functions and consider the arithmetic consequences of such an estimate in several different settings. We use this estimate to study mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros for Rankin-Selberg L-functions, the Chebotarev density theorem, and ℓ-torsion in class groups of number fields.
In the spirit of Linnik's original result, Kowalski and Michel [37,Theorem 5] proved a logfree zero density estimate for general families of automorphic L-functions in the conductor aspect. To describe their result, let A Q be the ring of adeles over Q, let d ≥ 1 be a fixed integer, and let A(d) be the set of cuspidal automorphic representations of GL d (A Q ) with unitary central character. We make the implicit assumption that the central character of each π ∈ A(d) is trivial on the positive reals; this discretizes A(d). For each π ∈ A(d), let L(s, π) = n≥1 a π (n) n s = p d j=1 (1 − α j,π (p)p −s ) −1 be the standard L-function associated to π, where p runs through the primes. Consider a finite set S(q) of distinct cuspidal automorphic representations π ∈ A(d) such that: (1) There exists some δ > 0 (depending at most on d) such that for each π ∈ S(q), each 1 ≤ j ≤ d, and each prime p, we have the bound |α j,π (p)| ≤ p 1/4−δ . (2) There exists a constant A > 0 such that for all π ∈ S(q), the conductor of π is O(q A ).
If σ ≥ 1 − M c and T is sufficiently small with respect to q, then (1.1) tells us that at most a vanishingly small proportion of low-lying zeros of the L-functions L(s, π) with π ∈ S(q) lie near s = 1. In many problems, such a result can serve as a powerful substitute for GRH. Until now, (1.1) appears to be the most flexible and robust zero density estimate for studying zeros of automorphic L-functions near s = 1.
Remark. Indeed, if π satisfies GRC, then Hypothesis 1.1 follows with lots to spare. Brumley [7, Theorem 1 and Corollary 2] proved that each π ∈ A(d) satisfies Hypothesis 1.1 when d ≤ 4 and gave sufficient conditions (strictly weaker than assuming GRC in full) under which π may satisfy Hypothesis 1.1 when d ≥ 5.
Remark. When π 0 ∈ A(1) is the trivial representation, whose corresponding L-function is the Riemann zeta function, Theorem 1.2 immediately recovers (1.1) (up to the quality of the coefficient of 1 − σ) with the added benefit of a significantly improved dependence on T . Theorem 1.2 is new for all other choices of π 0 , even if one assumes GRC in full.
Remark. For simplicity, we have made no attempt to optimize the exponent, but there is room for some noticeable improvement (especially if one assumes GRC). Obtaining such a numerical improvement was big component of the work in [60] (see Theorem 3.2).
Our proof of Theorem 1. The truth of Theorem 1.3 follows immediately from Theorem A.1, which we prove in the appendix. The bound in Theorem 1.3 along with (1.4) produces Theorem 1.2.
The bound Theorem 1.2 improves noticeably if π 0 satisfies GRC and there exists a primitive real Dirichlet character χ (mod q) with q ≤ 2Q such that L(s, χ) has real zero close to s = 1.
Page's theorem [13,Chapter 14] tells us that there exists an absolute and effective constant c 1 > 0 such that for every Q ≥ 3, there exists at most one modulus q ∈ (Q, 2Q] and at most one primitive real character χ (mod q) such that L(s, χ) has a real zero β χ with the property that β χ ≥ 1 − c 1 / log q. Moreover, such a zero β χ , which we call a Landau-Siegel zero, must be simple. If a primitive real character χ (mod q) with q ∈ (Q, 2Q] has an associated Landau-Siegel zero β χ , then Theorem 1.4 improves on Theorem 1.2. While it is well-known that Landau-Siegel zeros associated to real characters repel the zeros of Dirichlet L-functions from the point s = 1, Theorem 1.4 appears to be the first explicit instance in the literature where Landau-Siegel zeros associated to real characters repel zeros of high-degree L-functions. This adds to the growing literature on interesting consequences of the existence of Landau-Siegel zeros of Dirichlet L-functions [12,14,19,20,21,22,26]. Our proof of Theorem 1.2, which is noticeably different from that of (1.1), descends naturally from Gallagher's approach to log-free zero density estimates for Dirichlet L-functions [23]. Much like the classical approach to zero-free regions for L-functions, if L(s, π × π 0 ) has a zero ρ 0 such that |ρ 0 − (1 + it)| ≤ ε for some small ε > 0, then high derivatives of −L ′ /L(s, π × π 0 ) near s = 1 + ε + it will be large; this is made quantitative via the lower bound for power sums due to Sós and Turán [55]. Moreover, one can show that if these derivatives are large, then the mean value of a certain Dirichlet polynomial roughly of the shape P (t, π × π 0 ) = A<p<B a π×π 0 (p) log p p 1+it must also be large when t is close to Im(ρ 0 ). A new "pre-sifted" large sieve inequality (Proposition 5.1) in the spirit of Duke and Kowalski [15,Theorem 4] shows that the mean value of P (t, π × π 0 ) cannot be large for too many π ∈ F m (Q) simultaneously; Theorem 1.2 follows once this is made precise. The coefficients of P (t, π × π 0 ) are supported on large unramified primes, in which case a π×π 0 (p) = a π (p)a π 0 (p) by means of (1.2); this decisive identity facilitates the averaging over π ∈ F m (Q) while keeping π 0 fixed. We prove Theorem 1.4 similarly by simultaneously considering the twists L(s, π × π 0 ) and L(s, π × (π 0 ⊗ χ)) and exploiting the fact that if χ is a real primitive Dirichlet character with a Landau-Siegel zero, then χ behaves like the Möbius function. This approach contrasts with the method of proof for (1.1), which uses mollification to detect zeros and a mean value theorem involving Selberg's pseudo-characters to show that the aggregate contributions from the zeros of each L-function is small. It is unclear to the authors how one would modify the proof of (1.1) to incorporate a twist by π 0 while maintaining a log-free estimate.
In [58,Corollary 2.6], Soundararajan and the first author establish the first unconditional log-free zero density estimate for each individual Rankin-Selberg L-function L(s, π ×π 0 ) with an application to the weak subconvexity problem. The proof of [58, Corollary 2.6] relies on the same method of detecting zeros that we use here. Unfortunately, the means by which the proofs in [58] avoid appealing to a weak form of GRC (such as Hypothesis 1.1) appears to be incompatible with the process of averaging over π ∈ F m (Q). In particular, Hypothesis 1.1 appears to be indispensable in the proof of Proposition 5.1 unless #F m (Q) = 1, which is precisely the case considered in [58].

Arithmetic applications
2.1. Subconvexity and mass equdistribution. Let f be a Hecke-Maass newform for the congruence subgroup Γ 0 (q f ) ⊂ SL 2 (Z) with Laplace eigenvalue λ f and trivial central character. Define Let f 0 denote a fixed Hecke-Maass newform, and consider the L-functions L(s, f × f ) and The generalized Lindelöf hypothesis (which follows from GRH) predicts that for all ε > 0 and all f ∈ G (Q), we have the bounds I explicate the t and f 0 dependence: The so-called convexity bounds follow from the work of Heath-Brown [27]. Subconvexity bounds of the shape are not yet known; obtaining bounds of these sorts is a very active area of research which has some spectacular partial results (see [57,Theorem 1.1], for instance).
A standard calculation involving the approximate functional equation for Dirichlet Lfunctions and the large sieve shows that if Q is large, then for all except at most a density zero subset of the moduli q ≤ Q, we have the bound L(1/2, χ) ≪ ε q ε for all primitive Dirichlet characters χ (mod q). Similarly, a sufficiently strong analogue of the large sieve for automorphic forms will show that there exists a constant δ > 0 such that (2.2) holds for almost all f ∈ G (Q). The best candidate for such a large sieve is that of Duke and Kowalski [15,Theorem 4], but it falls short because the best unconditional bound toward GRC for Hecke-Maass newforms is not strong enough (though assuming GRC in full is not necessary). However, a straightforward application of Theorem 1.2 yields such an average result. Our interest in (2.2) is motivated by the quantum unique ergodicity conjecture. Lindenstrauss [44] and Soundararajan [56] proved that as f traverses the Hecke-Mass forms with q f = 1 and λ f → ∞, the L 2 mass of f equidistributes in Γ 0 (1) \ H with respect to the standard hyperbolic measure. This affirmatively resolved the quantum unique ergodicity conjecture of Rudnick and Sarnak [54] for the modular surface. More specifically, let where φ is a bounded measurable function on Γ 0 (1) \ H. It is now known that as f traverses the Hecke-Maass forms of eigenvalue λ f → ∞ with q f = 1, Unfortunately, the methods in [44,56] do not yield any information about the rate of convergence in (2.5). See [30,49,51] for an unconditional proof of (2.5) with an effective rate of convergence as f traverses the holomorphic cuspdial newforms of weight k f and level q f with k f q f → ∞; this proof relies heavily on the fact that GRC is known for such newforms. For work in the direction of establishing (2.5) for Hecke-Maass forms in q f -aspect when q f is large and prime, see [50].
We consider the problem of proving that for all except at most a density zero subset of f ∈ G (Q), one has (2.5) with a power-saving rate of convergence in the hybrid q f and λ f aspects. When f traverses the even Hecke-Maass forms with q f = 1, this follows from Zhao's computation of the quantum variance of the modular surface [65]. It is unclear to the authors whether one can adapt the proofs for the problem considered here.
Remark. By appealing to the extension of Watson's formula proved by Nelson, Pitale, and Saha (see [51]) and the calculations in [15,Page 11], one can extend the definition of G (Q) to allow q f to be any integer at the cost of allowing the exceptional set to be of size O φ (Q 1/2+ε ) in Corollary 2.2. The proof is entirely analogous.

2.2.
Rarity of Landau-Siegel zeros. Let F m (Q) be as in (1.3), and let π ∈ F m (Q)∩A(d). While GRH predicts that L(s, π) has no zero in the region Re(s) > 1/2, at present we know that L(s, π) has at most one zero in the region .
(See [32,Theorem 5.10].) If L(s, π) has a zero in this region, then π is self-dual (so the Dirichlet coefficients of L(s, π) are real), and the zero must be simple and real. We call such a zero a Landau-Siegel zero. Hoffstein and Ramakrishnan [29, Theorem A] proved that such Landau-Siegel zeros are quite rare. In particular, for some suitable effective constant c(m) > 0, there is at most one π ∈ F m (Q) such that L(s, π) has a real zero β satisfying β > 1 − c(m)/ log Q. This generalizes Page's theorem for Dirichlet characters. Moreover, it is known by the work of Hoffstein and Ramakrishnan [29, Theorem C] and Banks [2] that if m = 2 or 3, then no π ∈ F m (Q) has an L-function possessing a Landau-Siegel zero. The proof of [29, Theorem A] relies crucially on the cuspidality of the π ∈ F m (Q). The situation for Rankin-Selberg L-functions is much more difficult. Currently, an unconditional zero-free region (with at most one exceptional zero) roughly of the shape (2.7) exists for L(s, π × π 0 ) when at least one of π and π 0 is self-dual (see [31] for further discussion). Since it is not known in general whether L(s, π × π 0 ) factors into a product of L-functions associated to cuspidal automorphic representations (though this is expected), it is unclear how to unconditionally generalize [29, Theorem C] to establish the rarity of Landau-Siegel zeros for Rankin-Selberg L-functions. Despite these setbacks, one can still show that few Rankin-Selberg L-functions have a Landau-Siegel zero. Theorem 2.3. Assume the above notation. Let A > 0, and let S = S(A, Q, T, F m (Q)) be the set of all π ∈ F m (Q) such that L(s, π × π 0 ) has a zero in the region .
(ii) Let χ (mod q) be a primitive real Dirichlet character modulo q ≤ 2Q. Under the hypotheses of Theorem 1.
If there exists a sequence of primitive real characters χ (mod q) with q ∈ (Q, 2Q] such that (1 − β χ ) log Q → 0 as Q → ∞, then the size of the exceptional set in Theorem 2.3(ii) is zero once Q is sufficiently large relative to T . Therefore, under Hypothesis 1.1 for all cusp forms, the existence of a sequence of primitive real characters whose L-functions have a Landau-Siegel zero implies the nonexistence of Landau-Siegel zeros for all other Rankin-Selberg L-functions of comparable analytic conductor. This provides an interesting companion to another result of Hoffstein and Ramakrishnan [29,Theorem B] which roughly states that if all Rankin-Selberg L-functions factor into products of L-functions of cuspidal automorphic representations (as predicted by Langlands), then the only primitive L-functions over Q which could possibly admit a Landau-Siegel zero are those associated to primitive real Dirichlet characters.
Suppose that π 0 = π for all π ∈ F m (Q). By setting T = Q and A = log(C(π 0 )Q), it follows readily from Theorem 2.3(i) that apart from at most a few exceptional π in F m (Q), one can obtain strong approximations for L(1, π × π 0 ) as a short Euler product. See [11,24,40] for further discussion and applications of such approximations.
2.3. The Chebotarev density theorem in families. Let K be a number field of degree n = [K : Q] with D K = |disc(K/Q)| and Galois closure K over Q. Let G be isomorphic to the Galois group of K/Q, and let C be a conjugacy class of Gal( K/Q). Consider the prime counting function where the Artin symbol [ K/Q p ] denotes the conjugacy class of Frobenius automorphisms attached to the prime ideals of K which lie over p. The Chebotarev density theorem states that as x → ∞, where π(x) is the number of rational primes up to x. It follows from the work of Lagarias and Odlyzko [39, Theorem 1.1] that GRH for the Dedekind zeta function ζ K (s) implies We need the log log if we are to have an asymptotic. The least prime in Chebotarev being of size (log D) 2 is a consequence of smoothing. Unconditionally, refining a result of Lagarias and Odlyzko [39], it follows from work of Murty [48,Section 4] that where β 1 is a putative Landau-Siegel zero of ζ K (s). Recent work of the authors [61] shows for any A > 1, there exists B = B(A) > 1 such that For large x, (2.9) remains the strongest upper bound for E C and it is non-trivial in the absence of a Landau-Siegel zero. On the other hand, (2.10) exhibits a weaker estimate but for much smaller values of x. Nonetheless, even when ignoring the Landau-Siegel zero, both (2.9) and (2.10) fall far short of exhibiting non-trivial bounds for values of x commensurate in size with (2.8). Even establishing such bounds for x ≥ D o(1) K would be extremely desirable. Substantial progress has recently been made by Pierce, Turnage-Butterbaugh, and Wood [52] when K varies in certain families. They show that the ranges of x in (2.9) and (2.10) can be significantly improved for most K. We briefly summarize their results. Let G ∈ {C m , D p , S 3 , S 4 , A 4 }, where C m is a cyclic group of order m ≥ 2, S m is a symmetric group acting on m ≥ 2 elements, D p is a dihedral group of order 2p with p an odd prime, and A 4 is an alternating group acting on 4 elements. Let F (X) = F (X; G, n, R G ) denote the set of number fields K with [K : Q] = n and D K ≤ X such that Gal( K/Q) ∼ = G and each K satisfies a certain arithmetic restriction R G depending only on G. In particular, Every prime p that ramifies tamely in K has its inertia group generated by an element in the conjugacy class of reflections if G = D p , Every prime p that ramifies tamely in K has inertia group generated by an element in either {(1 As demonstrated in [52], there exists some constant a = a(G, n) ∈ (0, 1] such that, for all choices of G, n, and R G under consideration, #F (X) ≫ G,n X a .
With this setup in mind, let A ≥ 2 and η > 0. Pierce, Turnage-Butterbaugh, and Wood [52,Theorem 1.4] proved that there exist effective constants α = α(η, A, G, n) > 0 and ε = ε(G, n) > 0 such that for all fields K ∈ F (X) with at most O G,n (X −ε #F (X)) exceptions, one has (2.11) Notice (2.11) eliminates the Landau-Siegel zero and, most importantly, goes beyond the range of x in (2.10). Somewhat surprisingly, when a Landau-Siegel zero does not exist, the estimate for E C in (2.11) surpasses neither (2.9) nor (2.10) in their respective weaker ranges of x. (We have only collected their unconditional results; see [52, Section 2] for a discussion regarding degree n S n -and A n -fields with n ≥ 5.) The proofs in [52] rely decisively on (1.1), and the T -dependence in (1.1) inhibits their proof from achieving a result that is more commensurate with what GRH predicts in (2.8). Using Theorem 1.2 and Theorem 8.3, we improve both the range of x and quality of error term in (2.11). In particular, we obtain a range much closer to what GRH predicts with a power savings error term for small values of x.
Theorem 2.4. Let G be isomorphic to one of C m , S 3 , S 4 , D p , or A 4 ; let C ⊂ G be a conjugacy class; and let F (X) = F (X; G, n, R G ) be as above. There exist small positive constants η = η(G, n) and ε = ε(G, n) such that, for all fields K ∈ F (X) with at most O G,n (X −ε F (X)) exceptions, Remark. For a more uniform version of the error term in Theorem 2.4, see (8.15).

2.4.
Landau-Siegel zeros and torsion in class groups. Let us continue with the notation of Section 2.3. Let Cl K denote the ideal class group of a number field K. It is widely believed that if ℓ is a positive integer, then the ℓ-torsion subgroup Cl K [ℓ] is of size O ε,n,ℓ (D ε K ) for all ε > 0, while the trivial bound is O ε,ℓ,n (D  Since primes that split completely in K also split completely in K, the hypothesis (2.12) follows easily from (2.8), which is a consequence of GRH. It is a straightforward consequence of (2.11) that for any positive integer ℓ, all except at most a density zero subset of the fields K ∈ F (X; G, n, R G ) satisfy (2.12), and hence (2.13), unconditionally. This provides the first nontrivial upper bounds for |Cl K [ℓ]|, for all integers ℓ ≥ 1, applicable to infinite families of fields of arbitrarily large degree. This elegant application of (2.11) in [52] was achieved by exhibiting large zero-free regions for ζ K (s) for most fields K in a given family.
We proceed in a complementary direction using the zero repulsion phenomenon of a Landau-Siegel zero. If the Dedekind zeta function of a quadratic subfield Q( √ d) has a Landau-Siegel zero, then Theorem 1.4 implies that certain number fields K, whose Galois closure does not contain Q( √ d) as a subfield, possess GRH-quality bounds on ℓ-torsion in their class groups. (ii) Q( √ d) ∩ K = Q and log D K ≍ n,ε,ℓ log |d|. (iii) The Dirichlet L-function L(s, χ) has a real zero β χ = 1 − η χ / log d with η χ sufficiently small, depending only on n, ε, and ℓ. Then

Remarks.
(1) We emphasize that Theorem 2.5 is a pointwise bound, whereas the bounds in [52] hold as one averages over K. (2) Arthur and Clozel [1, page 223] proved that ζ K (s) is automorphic over Q when Gal( K/Q) is solvable. Therefore, by a well-known result of Feit and Thompson, hypothesis (i) on ζ K (s) is satisfied when |Gal( K/Q)| is odd.

Properties of L-functions
We recall some standard facts about L-functions arising from cuspidal automorphic representations and their Rankin-Selberg convolutions. Much of the material we present here can be found in [6, Section 1]. We refer the reader there for a more detailed overview.
3.1. Standard L-functions. Let d ≥ 1 be an integer, let A denote the ring of adeles over Q, and let A(d) be the set of all cuspidal automorphic representations of GL d (A) (up to equivalence). We consider each π = ⊗ p π p ∈ A(d) to be normalized so that π has unitary central character which is trivial on the positive reals; here, p ranges over the primes. We write π ∈ A(d) for the representation which is contragredient to π.
Let π = ⊗ p π p ∈ A(d), and let N π denote the conductor of π. The standard L-function L(s, π) associated to π is of the form The Euler product and Dirichlet series converge absolutely when Re(s) > 1. For each p, the local factor L(s, π p ) is given in the form a π (p j ) p js for suitable complex numbers α j,π (p). With this convention, we have α j,π (p) = 0 for all j whenever p ∤ N π , and it might be the case that α j,π (p) = 0 for some j when p | N π . At the archimedean place of Q, there are d complex Langlands parameters µ π (j) from which we define By the work of Rudnick for all j and p. The generalized Selberg eigenvalue conjecture and GRC assert that δ d = 0 for all d ≥ 1. For each p, Let r π denote the order of the pole of L(s, π) at s = 1 and κ π be the residue of L(s, π) at s = 1. The completed L-function is an entire function of order 1, and there exists a complex number W (π) of modulus 1 such that for all s ∈ C, Λ(s, π) = W (π)Λ(1 − s, π). On one hand, L(s, π) has a zero at each pole of L(s, π ∞ ); we call such a zero a trivial zero. On the other hand, since Λ(s, π) is entire of order 1, it has a Hadamard factorization Λ(s, π) = e aπ +bπs where ρ runs through the so-called nontrivial zeros of L(s, π). Finally, we define the analytic conductor of π to be . The Rankin-Selberg L-function L(s, π × π ′ ) associated to π and π ′ is of the form The Euler product and Dirichlet series converge absolutely when Re(s) > 1. For each p, the local factor L(s, π p ) is given in the form for suitable complex numbers α j,j ′ ,π×π ′ (p). With δ d as in (3.1), we have the pointwise bound If p ∤ N π N π ′ , then we have the equality of sets At the archimedean place of Q, there are d ′ d complex Langlands parameters µ π×π ′ (j, j ′ ) from which we define These parameters satsify and satisfy the pointwise bound Let r π×π ′ be the order of the pole of L(s, π × π ′ ) at s = 1 and let κ π×π ′ be the residue of L(s, π × π ′ ) at s = 1. By our normalization for π and π ′ , we have that r π×π ′ = 1 if and only if π = π ′ ; otherwise, r π×π ′ = 0 and hence κ π×π ′ = 0. The function is entire of order 1, and there exists a complex number W (π × π ′ ) of modulus 1 such that Λ(s, π × π ′ ) satisfies the functional equation On one hand, L(s, π × π ′ ) has a zero at each pole of L(s, π ∞ × π ′ ∞ ); we call such a zero a trivial zero. On the other hand, since Λ(s, π × π ′ ) is entire of order 1, it has a Hadamard factorization where ρ runs through the so-called nontrivial zeros of L(s, π × π ′ ). As with L(s, π), we define the analytic conductor of π ⊗ π ′ to be It will be important to be able to decouple the dependencies of C(π × π ′ , t) on π, π ′ , and t.
When π, π 0 ∈ A(1) and π 0 is trivial, Proposition 4.1 reduces to a result of Weiss [63, Proposition 4.2]; we follow Weiss's proof with the modifications which follow [42,58] to allow for more general choices of π and π 0 . Relative to the ideas in [42,58,63], there are three novelties here. First, we exploit the existence of an exceptional zero of a Dirichlet L-function in the zero-detection process for L(s, π × π 0 ), which generalizes [63,Proposition 4.2]. Second, we use Hypothesis 1.1 for both π and π 0 instead of assuming that at least one of π and π 0 satisfies GRC as in [42] so that, unlike the approach in [58], the Dirichlet polynomial can be supported on primes. Third, much like [58,Section 4], the proof here makes explicit some of the effective constants in [42,63], which we believe makes the proof a bit easier to read.
Lemma 4.4. If π ∈ A(d) satisfies Hypothesis 1.1, y > C(π), and η is as in (4.4), then Proof. We first bound the contribution to the sum in (4.6) from the n which share a prime factor with N π separately. Note that O(log y) primes divide N π as y > C(π) ≥ N π . Thus by (3.4) and (4.1) applied to the ramified prime, we have 2≤r≤20000 log y y 1/r ≤p≤y 12000/r p|Nπ If p ∤ N π , then λ π× π (p r ) = |λ π (p r )| 2 . From (4.1), we see that Note that β p ≤ p −2/(d 2 +1) by (3.2). Thus the contribution to the sum in (4.6) arising from the n which are coprime to N π is Subject to Hypothesis 1.1, we will prove that (4.7) uniformly for all 2 ≤ R ≤ 20000 log y, which suffices to prove the lemma.
The inner sum is geometric, so We decompose the sum according to whether p is greater than 2 d 2 (in which case 1 −β p p −η ≥ 1/2) or not. The contribution from the latter range to the sum is O d (1), so we have The above display is of size 2ε log y + O d (1) by Hypothesis 1.1, which establishes (4.7).
we can apply Lemma 4.5 to the sums over zeros in (4.10) and find that if the implied constant in (4.11) is sufficiently large, then for some k ∈ [K, 2K], It follows from the calculations on [63, pages 80-81] that Therefore, for some k ∈ [K, 2K] with K given by (4.11), we have the lower bound On the other hand, since η > 0, we can use the absolute convergence of the Dirichlet series which defines F (s) to directly compute where j k (u) = (k!) −1 u k e −u . Let A 1 = exp(K/(300η)) and A 2 = exp(40K/η). Suppressing summands, we write the right hand side of (4.13) as First, we bound the contribution from n / ∈ [A 1 , A 2 ]. Since k! ≥ (k/e) k , we find from a small numerical calculation [58, Proof of Lemma 4.3] that By (4.15), ). Using (4.11), we see that the contribution from n / This estimate and (4.3) imply that By Lemma 4.4 and (4.9), the above display is Finally, we estimate the contribution from the primes p ∈ [A 1 , A 2 ]. Summation by parts gives us the identity Much like the above calculations, we use Lemma 4.2 to deduce that the sum over If K is given by (4.11), then the condition p ∈ [A 1 , A 2 ] implies that p ∤ N π N π 0 q. Therefore, by (3.5) and (4.1), We collect our estimates for the three sums in (4.14) to find that for all k ∈ [K, 2K] with K given by (4.5), (4.17) We enlarge K according to (4.5), which we are free to do. If k ∈ [K, 2K] and the implied constant in (4.5) is sufficiently large, then O m,m 0 (k(110) −k ) ≤ 1 4 (100) −k−1 . Therefore, it follows from (4.12) and (4.17) that if L(s, π × π 0 ) has a zero ρ 0 = β χ which satisfies |ρ 0 − (1 + iτ )| ≤ η, then with K given by (4.5), we have the bound We square both sides and apply the Cauchy-Schwarz inequality to obtain the bound

A new large sieve inequality
We will generalize the large sieve for Dirichlet coefficients of automorphic representations due to Duke and Kowalski [15,Theorem 4]. As observed by Brumley [7], one can adjust their proof to show that if F m (Q) satisfies Hypothesis 1.1 and Q, x ≥ 2, then where b(n) is any complex-valued function supported on the integers. We require two modifications to (5.1). First, we need to take sums over n in intervals of length x/T , where T is arbitrarily large. Second, we need a variant of (5.1) which applies with more sensitivity to sequences b(n) supported on the primes. We establish a "pre-sifted" large sieve inequality over short intervals for families of automorphic representations which satisfy Hypothesis 1.1. We anticipate that this will be useful in contexts beyond this paper. In what follows, we define P − (n) to be the least prime dividing a positive integer n; we set P − (1) = ∞ by convention.
Proposition 5.1. Let b(n) be a complex-valued function supported on the integers, and suppose that each π ∈ F m (Q) (see (1.3)) satisfies Hypothesis 1.1. If Q ≥ 3, T ≥ 1, x > 0, and z ≫ m Q 6m with a sufficiently large implied constant, then for every ε > 0, We call the Dirichlet series the naïve Rankin-Selberg L-function. We access the Dirichlet coefficients of L RS (s, π × π ′ ) by relating L RS (s, π × π ′ ) to L(s, π × π ′ ). In order to accomplish this, we use Hypothesis 1.1 and the following result of Brumley (see the proof of [7, Corollary 3]).

Preliminary estimates.
Let π, π ′ ∈ F m (Q), and assume throughout this subsection that both π and π ′ satisfy Hypothesis 1.1. Let and let d ≥ 1 be a square-free integer which is coprime to N π N π ′ . We will consider the Dirichlet series given by A bound for g RS d (s, π × π ′ ) follows readily from (3.4). Lemma 5.3. Let d ≥ 1 be square-free and π, π ′ ∈ F m (Q). In the region The lemma now follows from the well-known bound ω(n) ≪ (log log n) −1 log n, where ω(n) is the number of distinct prime factors of n.
Proof. First, we establish the bound for every ε > 0. By the work of Li [43, Theorem 2], we know that for some constant c m > 0 depending at most on m, By replacing π ′ with π ′ ⊗ | det | −it in the proof of (5.5) (which does not change the proof substantially), we obtain The refined version of the convexity bound for L-functions proved by Heath-Brown in [27] yields Hence, by (5.6), Thus (5.4) follows from (3.7), (5.6), (5.8), and an application of the Phragmén-Lindelöf principle. We see from (3.4) and the bound ω(n) ≪ (log log n) −1 log n that for every ε > 0, one has the bound With this bound in hand, the lemma follows from Lemma 5.2, Lemma 5.3, and (5.4).
Fix a smooth function φ whose support is a compact subset of (−2, 2). Let Thus φ(s) is entire, and integrating by parts several times yields the bound any integer k ≥ 0. Let T ≥ 1; by Fourier inversion, one has the identity for any x > 0 and any c ∈ R.

Proof of Proposition 5.1. It suffices to prove the bound
|b π | 2 (5.10) for any sequence of complex numbers {b π } π∈Fm(Q) with the convention that a π (n) = 0 when (n, N π ) > 1, where Q ≥ 3, T ≥ 1, x > 0, and z ≫ m Q 6m . Indeed, with (5.10) in hand, it follows from a standard application of the duality principle that again with the convention that a π (n) = 0 when (n, N π ) > 1 and with Q, T , x, and z as before. Since z > Q by hypothesis and Q ≥ N π for all π ∈ F m (Q), the condition P − (n) > z implies that (n, N π ) = 1 for all n ∈ (x, xe 1/T ]. The proposition now follows. To bound (5.10), we choose a compactly supported, infinitely differentiable function φ such that φ(t) ≥ 1 for t ∈ [0, 1] and φ(t) ≥ 0 otherwise. Then φ(T log n x ) is a pointwise upper bound for the indicator function of the interval (x, xe 1/T ]. If w z is any function such that w z (n) ≥ 1 if P − (n) > z and w z (n) ≥ 0 otherwise, then (5.11) We expand the square, swap the order of summation, and apply [15,Lemma 1] so that the righthand side of (5.11) equals We now choose w z (n) as in the Selberg sieve. Let π 1 ∈ F m (Q) be a representation which achieves the maximum in (5.12). Let g(d) = g d (1, π 1 × π 1 ), and define Let ρ d be a real-valued function such that Upon expanding the square and swapping the order of summation, (5.12) equals Our convention that a π (n) = 0 if (n, N π ) > 1 for each π ∈ F m (Q) means that the above display is bounded by We use Lemma 5.5 along with condition (3) to bound (5.13) by |b π | 2 . (5.14) By proceeding as in the formulation of the Selberg sieve in [18,Theorem 7.1], we find that there exists a choice of ρ d satisfying conditions (1)-(3) such that Note that κ π× π , H(1, π 1 × π 1 ), and p|Nπ 1 L RS (1, (π 1 ) p ×( π 1 ) p ) −1 are each positive. Therefore, by Lemma 5.5 and the upper bound φ(1/T ) ≪ 1 from (5.9), we have that if z ≫ m Q 6m with a sufficiently large implied constant, then the main term in (5.14) is bounded by Proof. A result of Gallagher [23, Theorem 1] states that for any sequence of complex numbers a n and any T ≥ 1, we have Assume z ≥ c m Q 6m with c m sufficiently large. If b(n) is as in Proposition 5.1, then the above result with a n = b(n)a π (n) yields the bound We apply Proposition 5.1 and bound the right hand side of the above display by Choose y such that y ≥ c m (C(π 0 )QT #F m (Q)) 32(m 0 m) 3 and y > z, and choose b(n) to be supported on the primes p > y. Then the above display is Choose z = y 1/(5m 2 ) so, by our assumption on y, we indeed have z ≥ c m Q 6m . It follows that otherwise.

The Chebotarev density theorem in families
The goal of this section is to prove Theorem 2.4. Let L/Q be a Galois extension of number fields. We begin by establishing a flexible variant of the Chebotarev density theorem. Given any zero-free region for the Dedekind zeta function ζ L (s), we would like to compute an asymptotic expression for π C (x, L/Q) with an error term depending on the zero-free region in an explicit form. Remarks.
(1) The existence of this abelian subgroup H is a mild condition for our purposes. In the special case C = {1}, one can take H = {1} and this follows unconditionally from the Aramata-Brauer theorem as L H = L is Galois over Q. For an arbitrary conjugacy class C, one can take H = g to be the cyclic subgroup generated by some element g ∈ C in which case this assumption follows easily from the strong Artin conjecture for ζ L (s) over Q. The strong Artin conjecture is known for all examples under consideration in Theorem 2.4.
(2) An analogous result holds for any Galois extension L/F with π(x) replaced by the number of prime ideals of F up to x and ζ Q (s) replaced by ζ F (s). We restrict to F = Q for simplicity and with Theorem 2.4 in mind.
Proof. For the proof, we will borrow heavily from results recorded in [61] and will therefore remain consistent with the notation therein. Let g ∈ H ∩ C be arbitrary and set C H = {g}.
where χ runs over all the (Hecke) characters of the dual group H. By [61,Lemma 4.3], the bound ε ≥ x −1/4 from (8.2), and the bounds n L ≪ log D L ≤ x 1/4 , it follows that By standard arguments using Mellin inversion, one can verify that All that remains is to consider the error term in (8.4). By [61,Lemma 4.4] and the assumption log D L ≤ x 1/4 , the zeros ρ with |ρ| ≤ 1/4 have negligible contribution; namely, Write ρ = β + iγ for each non-trivial zero ρ. By (8.1), one can see that Thus, [61, Lemma 2.2(iv)] and (8.2) imply that, for |ρ| ≥ 1/4, Summing over all such zeros, it follows that Applying a standard estimate for the zeros of the Dedekind zeta function [61, Lemma 2.5] and Minkowski's bound n L ≪ log D L , we see that the above expression is Substituting (8.7), (8.6), and (8.4) into (8.3), we conclude that By (8.1), one can verify that η(y) is an increasing function of y and also η(x 1/2 ) ≥ 1 2 η(x). With these observations and the assumption log D L ≤ x 1/4 , we conclude that .
Assuming a strong zero-free region for the Dedekind zeta function, we arrive at a natural form of the Chebotarev density theorem Proof. By Proposition 8.1 and Lemma 8.2, it remains to compute η(x) for and If η(x) = η 1 (x), then η(x) ≥ δ log x. Otherwise, we may assume η(x) = η 2 (x). Arguing as in [61,Lemma 4.6], the expression c 5 log x log D L +n L u + u is positive for u ≥ 0 and is globally minimized in this interval at u = max{0, Since one always has the lower bound η 2 (x) ≥ log T ≥ 24 log D L , we see in all cases that We conclude this section with the proof of Theorem 2.4.
Proof of Theorem 2.4. Recall F (X) = F (X; n, G, R G ) is a family of number fields over Q whose Galois closure has Galois group is isomorphic to G, where G is a fixed transitive subgroup of S n equal to one of C n , S 3 , S 4 , D p or A 4 . Let K ∈ F (X) and recall K/Q is the Galois closure of K over Q. For Re(s) > 1, where ρ runs over the non-trivial irreducible Artin representations of G. In all cases under consideration, the strong Artin conjecture is known for all of the non-trivial Artin representations ρ of G. That is, L(s, ρ, K/Q) = L(s, π) for some cuspidal automorphic representation π = π ρ of GL d (A Q ) with d equal to the degree of ρ. Observe that d is bounded by m, where m = m(G) is the maximum degree of the irreducible representations of G. The map has image A (X) = A (X; G, n, R G ), the set of automorphic representations π obtained this way from F (X). Let M(X) = M(X; G, n, R G ) be the maximum size of the fibres of the map in (8.10). As shown in [52], where the maximum runs over all number fields F = Q. Since our notation differs with theirs, we explain (8.11) for the sake of clarity. Fix some π ∈ A (X). = F for some number field F . Note that F = Q since the representations ρ 1 , ρ 2 are non-trivial. Hence, the size of the fibre above π ∈ A (X) in (8.10) equals #{K ∈ F (X) : Q ⊂ F ⊆ K} for some number field F = Q, implicitly depending on π. This implies (8.11).
In light of (8.11), it follows from [52,Proposition 7.9] and [52, Theorem 7.1] that there exists a sufficiently small ε = ε(n, G) > 0 such that This result is one of the key innovations of [52]. Now, we verify the assumptions of Theorem 1.2 with π 0 ∈ A(1) taken to be the trivial representation. Take m = m(G) to be the maximum degree of the irreducible representations of G, Q = X |G|/2 , and F m (Q) = A (X). By (8.9) and (8.10), each π ∈ F m (Q) satisfies deg(π) ≤ m and C(π) ≤ D K for some K ∈ F (X).
Each exceptional π corresponds to at most M(X) exceptional fields K ∈ F (X). Throwing out all of these exceptional fields, it follows by (8.12) that ζ K (s)/ζ Q (s) is zero-free in the region (8.14) for all K ∈ F (X) with at most O n,G,ε (X −ε #F (X)) exceptions. Now, let K ∈ F (X) be a non-exceptional field. By Theorem 8.3, we have that Note we used that D K ≤ Q to express E(x) in terms of D K instead of Q. Choose η = δ/8.

Landau-Siegel zeros and torsion in class groups
This section is dedicated to the proof of Theorem 2.5. The first ingredient is a lemma due to Ellenberg-Venkatesh [16,Lemma 2.3]. It establishes a connection between the existence of small split primes and bounds for the class group. To make use of Lemma 9.1, we require a proposition relating low-lying zero free regions to the existence of small primes with a given splitting behaviour.
Proposition 9.2. Let L/Q be a Galois extension of number fields and let 0 < ε < δ/2 be arbitrary. Suppose ζ L (s) has no zeros in the region where H δ,ε ≥ 1 is sufficiently large. Then, for any conjugacy class C ⊆ G , Proof. This essentially follows from the arguments found in [64]. We will outline the proof here and borrow heavily from [64], so we will remain as consistent as possible with the notation therein. In particular, set L = log D L . Select f as in [64, Lemma 2.6] with ℓ = 2, B = δ, and A = ε/4. Then • For s = σ + it ∈ R with σ < 1 amd t ∈ R, we have: Furthermore, F (0) = 1.
We will use these properties frequently and often without mention. Define where, for primes p unramified in L, 1 C (p) = 1 if [ L/Q p ] = C and 0 otherwise. By the properties of f , one can verify that

Now, from the proof of [64, Lemma 4.1], we have that
where ψ runs over the irreducible Artin characters of Gal(L/Q). Using standard class field theory arguments (see [64,Section 4.2]), one can shift the contour as in [64,Lemma 4.2] with T ⋆ = 1. This yields where ρ runs over all non-trivial zeros of ζ L (s) satisfying |Im(ρ)| ≤ 1. We apply [64,Lemma 4.3] (with J = 1, T 1 = 1, and R 1 = H δ,ε in their notation) to deduce that By assumption (9.1), the remaining sum over zeros is empty. Combining these estimates with (9.3) implies that since H δ,ε is sufficiently large. Substituting this lower bound into (9.2) yields the result.
Proof of Theorem 2.5. Recall K is a number field of degree n and K is its Galois closure over Q. By assumption, ζ K (s) = L(s, π) for some automorphic representation π of GL m (A Q ) with m = [ K : Q] ≤ n!. Clearly, L(s, π) satisfies GRC. Let and let H δ,ε ≥ 1 be sufficiently large. From the estimate D , one can see that δ < 1 and δ is bounded away from zero uniformly in terms of n, ℓ, and ε. Thus, when a quantity depends on δ (such as H δ,ε ), we may replace this dependence with n, ℓ, and ε. In particular, we may treat δ as independent of D K and D K .
We note that while ζ K (s) does not directly satisfy the hypotheses of Theorem 1.4 (as ζ K (s) is not the L-function of a cuspidal automorphic representation Q, and it only conjecturally factors into a product of such L-functions), it has an analytic continuation and functional equation just as described in Section 3. The only part of the proof of Theorem 1.4 which relies on cuspidality is in the use of Proposition 5.1. However, since we are considering a single L-function here instead of several, the use of Proposition 5.1 can be replaced with the field-uniform analogue of the Brun-Titchmarsh theorem proved in [25,Proposition 2].
We apply Theorem 1.4 with π 0 trivial, F m (Q) = {π}, T = 1, and σ = 1 − Since we have log Q ≍ n,ε,ℓ log D K ≍ n,ε,ℓ log d, the bound m ≤ n! and β χ = 1 − ηχ log d , we have that N π (1 − H δ,ε log D K , 100) ≪ n,ℓ,ε η χ . As η χ sufficiently small depending only on n, ℓ, ε, it follows that ζ K (s) has no zeros in the region Thus, by Proposition 9.2, there are M rational primes p ≤ D δ K = D , provided D K is sufficiently large depending on ε, n, and ℓ. The result now follows from an application of Lemma 9.1 and rescaling ε appropriately.
Appendix A. Explicit upper bound on the universal family for GL n Let F be a number field of degree d over Q and discriminant D and let n ≥ 1 be an integer. Let A cusp denote the set of unitary cuspidal automorphic representations π of GL n (A F ), with normalized central character, ordered by analytic conductor C(π). We recall that C(π) = D n N π k p i, where N π = Norm(q π ) is the arithmetic conductor and k p i the archimedean conductor, as in (3.3). Note the factor of the discriminant, which arises naturally in the functional equation for the standard L-function of π.
For Q ≥ 1 let F (Q) = {π ∈ A cusp : C(π) ≤ Q}. We present an argument, due to Venkatesh [62] and based on results in [6], to deduce a polynomial upper bound on the cardinality |F (Q)|. We can in fact make this polynomial bound explicit, using subsequent refinements of loc. cit., as in the following Theorem A.1. We have, for all fixed ε > 0, |F (Q)| ≪ d,n,ε (D −n 2 Q 2n ) 1+ε .
Remark. As we consider d and n as being fixed, we shall henceforth systematically suppress the dependence of implied constants on n and d in the notation.
Remark. The expected value of the exponent of Q in Theorem A.1 is n + 1, and indeed this was shown (with an asymptotic) in [8], with one caveat: for n ≥ 3 the authors restrict to the subfamily of F (Q) consisting of Maass forms. This restriction is fortunate, in a way, since it provides an occasion for this appendix, which has sat for a long time in a drawer (or inbox) and whose methods are quite different. While Theorem A.1 says nothing about existence, and the upper bound is not sharp, we believe that the proof itself is of sufficient interest to merit circulation.
Remark. The results of [8] make no claim of uniformity in the number field F . (In fact one should note the difference in the notational conventions between these two papers: in [8], the analytic conductor, denoted by Q(π) there, does not include the factor of the discriminant.) The upper bound in Theorem A.1 is, however, uniform in D, making this perhaps the most novel aspect of the result.
The proof of Theorem A.1 combines two ingredients: Rankin-Selberg theory and sphere packing bounds in large dimensions. It is natural to ask what effect assuming standard conjectures on these L-functions would have on the quality of the resulting bound. For example, a similar argument to the one we present here was used in [17] to count ℓ-adic sheaves of bounded complexity. In that article, Deligne's proof of the Riemann hypothesis over finite fields is used to show that certain trace functions form a quasi-orthogonal system with small enough angular separation to deduce a polynomial upper bound. We show that the exponent 2n can be improved to n + 1 under standard conjectures, demonstrating the strength of the method of proof.
Theorem A.2. Denote by F χ (Q) the subfamily of F (Q) having fixed central character χ. Assume the Ramanujan conjecture and the Riemann hypothesis for Rankin-Selberg Lfunctions. Then |F χ (Q)| ≪ ε (D −n 2 /2 Q n ) 1+ε , and Remark. Note that, by the results in [8], the exponent of Q in Theorem A.2 is sharp, up to the ε. Moreover, the D dependence here and that of the main term of the asymptotic given in [8] are in agreement.
Remark. The method of proof of Theorems A.1 and A.2 is sensitive to any loss of information incurred in the application of the Bushnell-Henniart bounds [9]. Recall that the main result in loc. cit. provides upper bounds for the Rankin-Selberg Artin exponent Ar(π v ×π ′ v ) at finite places v in terms the standard Artin exponents Ar(π v ) and Ar(π ′ v ), and the integers n, n ′ , where π v and π ′ v are smooth irreducible representations of GL n (F v ) and GL n ′ (F v ), respectively.
While the bounds in loc. cit. are sharp in general, we apply them under additional hypotheses on π v and π ′ v . Namely, in the course of the proof, we assume that (1) the dimensions n = n ′ are the same, (2) the Artin exponents a = Ar(π) = Ar(π ′ ) are the same, (3) the central characters are the same, say equal to χ. Under the assumptions (1) and (2)  This decomposition allows one to prove, in principle, refined bounds for the cardinality of these subfamilies, since the Bushnell-Henniart bounds [9] can often be improved under such assumptions. For example, if the combinatorial data that one takes is "trivial", in the sense that it corresponds to π v and π ′ v supercuspidal on GL n , then (keeping the assumptions (2) and (3) of the previous remark) one can use the bound Ar(π ×π ′ ) ≤ na of [10, Corollary C], which is, in general, far better than the general bound of (2n − 2)a cited above. In this way one can show that, under Ramanujan and Riemann as in Theorem A.2, the subfamily of F (Q) consisting of π which (1) are supercuspidal at all the places at which they ramify, (2) have archimedean component lying in some fixed compact of the unitary dual, has cardinality O(Q n 2 +2 ) (ignoring the discriminant dependence). This bound is surprisingly strong, and no trace formula was used to derive it. We have not found this type of interplay between conductor dropping phenomenon and improved bounds on dimension counts of automorphic forms elsewhere in the literature.
A.1. Idea of proof. We present here the basic argument to prove Theorem A.1. We shall later need to modify the presentation to obtain the best possible exponent.
Let q be an integral ideal of O F . Let χ be a character of A × f of conductor q, where A f is the ring of finite adeles. Let Here, q π f is the conductor of π f and χ π f is the central character of π f . Then The argument we sketch below provides a bound on |F q,χ (Q)| of the form Executing double sum over pairs (q, χ), this would produce a bound of O ε ((D −n 2 Q 2n+n 2 ) 1+ε ). We will later show how to remove the n 2 to establish Theorem A.1, as well as the sharp conditional bounds in Theorem A.2.
A.1.1. Mapping A cusp (q, χ) to a Hermitian space. We begin by describing a way to map A cusp (q, χ) to a Hermitian space, whose inner product can be understood in terms of Rankin-Selberg L-functions. The reader is encouraged to read ahead to the next subsection describing the Dirichlet coefficients of these L-functions for a motivation of the following constructions.
Recall that a partition µ = (µ i ) is a sequence of non-increasing non-negative integers µ 1 ≥ µ 2 ≥ · · · with only finitely many non-zero entries. Write P for the set of all partitions. The length of µ ∈ P, denoted ℓ(µ), is the number of its non-zero entries. Write for the partitions of length at most ℓ. Finally, for µ = (µ i ) ∈ P, write |µ| = i µ i . For an integer r, let P ℓ (r) = {µ ∈ P ℓ : |µ| = r}; this is the empty set when r is negative.
Let S be a finite set of finite places. Let I S denote the set of integral ideals of O F supported outside of S. When S is empty we abbreviate this to I for the set of all integral ideals. Given an n = p p rp ∈ I we write P n−1 (n) for the set of sequences µ = (µ p ) p of partitions such that µ p ∈ P n−1 (r p ). A P n−1 -decorated prime-to-S ideal is a pair (n, µ), where n ∈ I S and µ ∈ P n−1 (n). Let I S denote the set of P n−1 -decorated prime-to-S ideals. We have a map I S → I S , (n, µ) → n, where we forget the decoration and take the underlying ideal. Observe that several (n, µ) can have the same underlying ideal n. We shall sometimes writeñ for a P n−1 -decorated ideal with underlying ideal n.
For a parameter X > 1, let I S (X) = {ñ ∈ I S : Norm(n) ≤ X}; this is the set of pairs (n, µ) with Norm(n) ≤ X and µ ∈ P n−1 (n). Let V S (X) be the vector space of complex valued functions on I S (X). Endow V S (X) with the standard scalar product For an integral ideal q of O F , with support S, we shall map A cusp (q, χ) to V S (X) in the following way.
We note that if n = 2 and π f has trivial central character, then there is no decoration µ and the (A.2) just recovers the Hecke eigenvalue of π at n. In fact, more generally, when n ≥ 2 and π f has trivial central character, if we take µ = (µ p ) p to satisfy µ p = (r p , 0, . . .), then we once again recover the Hecke eigenvalue at n = p p rp .
Let f : R → R be a non-negative smooth function supported in 1 2 , 1 and having Lebesque integral 1. Write f (Norm(nm n )/X).
For every π ∈ A cusp (q, χ) we define a vector v S π ∈ V S (X) by the rule v S π : (n, µ) → F S X (n)a π (n, µ). Note that for Norm(n) > X we have F S X (n) = 0; in this way the functionñ → v S π (ñ) can indeed be viewed as an element of V S (X).
A.1.2. Relation to Rankin-Selberg L-functions. We now recall the description of the Rankin-Selberg Dirichlet coefficients. This will clarify the choice of map π → v S π and the inner product we put on V S (X).
The sum on k is finite, going up to the integer part of r/n. Note that, in the above expression, we have used the fact that χ π f = χ π ′ f = χ; this explains why we've decomposed according to central character in (A.1). Thus, for (n, S) = 1, we have a π×π ′ (n) = a π (n, µ)a π ′ (n, µ) We recognize this as v S π , v S π ′ . On the other hand, if we let be the Mellin transform of f , then by the Mellin inversion formula one has This allows us to read off the orthogonality properties of v S π and v S π ′ in terms of the analytic information of L S (s, π ×π ′ ).
be the projection of the vector v S π to the unit sphere in V S (X). The idea behind the proof of Theorem A.1 is to show that, for X large relative to Q, (1) the map F q,χ (Q) → V S given by π → v S π is injective; (2) when π, π ′ ∈ F q,χ (Q) are distinct, the vectors u S π and u S π ′ are quasi-orthogonal; (3) there cannot be too many such quasi-orthogonal vectors. Moreover, each of these steps will be seen to be quantifiable, polynomially in Q.
There is only one problem with this approach: we have thrown out the information at ramified primes. While this allows for a simpler presentation, the price to pay is a weaker bound in Theorem A.1. Indeed one obtains in this way the exponent 2n + n 2 + ε in the parameter Q, with or without assuming the Ramanujan conjecture and the Riemann hypothesis. See Remark A.4.2 for more details on the source of this loss by a power of n 2 .
To obtain the unconditional bound of Theorem A.1 (as well as the conditional bound of Theorem A.2, which is sharp up to ε), we shall need to take into account the information at ramified primes. To adapt the above argument along these lines, one must explicate the Rankin-Selberg coefficients at ramified primes, which has been done by Brumley in [58,Appendix]. In particular, we shall see in §A.2 that the "combinatorial distance to supercuspidal" of π S = ⊗ p∈S π p governs the shape of the ramified Rankin-Selberg coefficients. Then, in §A.3, we further decompose F q,χ (Q) according to this data. After an appropriate enrichening of the space V S (X) to take into account this information, we then execute the above three steps.
A.2. Rankin-Selberg theory. We now recall some of the basic local and global properties of the Rankin-Selberg L-function that we shall need in the proof of Theorem A.1.
A.2.1. Induction data. Let v be a finite place of F associated with a prime ideal p of O F . Let q v be the cardinality of the residue field. Let π v be an irreducible unitary generic representation of GL n (F v ).
Recall that by the Bernstein-Zelevinsky description of admissible dual, we may associate with π v (see [58, §A.2]) the following combinatorial data: (C1) a standard Levi subgroup M ≃ GL n 1 × · · · × GL nr of GL n ; (C2) a partition J = [J 1 , . . . , J A ] of the set {1, . . . , r}; (C3) an integer vector d = (d 1 , . . . , d r ) ∈ N r , where d j | n j , such that m j = n j /d j is constant (say equal to m a ) along j ∈ J a ; (C4) an integer vector e = (e 1 , . . . , e A ) ∈ N A , where each e a | n; the following analytic data: (A1) real numbers σ 1 ≥ · · · ≥ σ r ; (A2) real numbers t 1 , . . . , t r ; encoded in the complex numbers as well as the following arithmetic data: where J ν a = {j ∈ J a : n j ≥ ν}. We expand the expression (A.5) into the local Dirichlet series, which we again denote by a π×π ′ (p r ). We shall now describe these in terms of the analytic data z j , similarly to the unramified setting of §A.1.
We now furthermore assume that the central characters of π v and π ′ v coincide. We fix a and ν in (A.5) and expand the product over j and k. We obtain a π×π ′ (p ear ; ν, a)X ear .
The convexity bound of Li [43] (see also [7] for the cases n = 3, 4) states that We have the factorization C(π ×π ′ , s) = D n 2 N π×π ′ K π×π ′ (s). For π f and π ′ f of conductor q, whose central characters are equal up to an unramified twist, Theorem B.1 of Appendix B implies that Moreover, the bounds [41, Lemma A.2] imply We deduce that, for π, π ′ ∈ F q,χ (Q), we have The function L(s, π ×π ′ ) is regular at s = 1 if and only if π ′ = π. In the case where π ′ = π, we have a lower bound of polynomial type on the residue at s = 1. Indeed, [6,Theorem 3] establishes the existence of an A > 0 such that Remark. In [7] it is shown that Res Recall the set I S of P n−1 -decorated prime-to-S ideals from §A.1. We shall now enrich I S at the places in S to account for the combinatorial information C . We shall call a (P n−1 , C )decorated ideal a triple (n, µ, (a, ν)), where n ∈ I is an integral ideal, µ ∈ P n−1 (n), and (a, ν) ∈ Index(C ). We shall generally write this as (n, µ; a, ν). The set of such triples shall be denoted I S . We have a map I S → I, (n, µ; a, ν) → n, where we forget the decorations and take the underlying ideal n. We sometimes writeñ for an element in I S with underlying ideal n. Let I S (X) denote the set ofñ ∈ I S with Norm(n) ≤ X.
Let V S (X) be the vector space of complex valued functions on I S . Endow V S (X) with the standard scalar product We shall map A cusp (q, χ, C ) to V S (X) by sending π ∈ A cusp (q, χ, C ) to the vector v π ∈ V S (X) given by the formula v π (ñ) = F X (n)a π (ñ), where f : R → R is as in §A.1 and The above enrichment allows us to identify the inner product v π , v π ′ in terms of the full finite part Rankin-Selberg L-function. Indeed, by (A.10) and Mellin inversion we have The above formula is the culmination of the combinatorial explication of the Rankin-Selberg L-functions in §A.1-A.3. It is the basis of the following section.
A.4. Executing steps (1) and (2). We now execute the first two steps of the proof outline in §A.1, using the facts we collected from Rankin-Selberg theory in §A.2.4.
A.4.1. First step. We begin by establishing the following result.
Proof. Indeed, [6,Theorem 7] shows the existence of a B > 0 such that when X ≫ Q B any pair (π, π ′ ) ∈ F q (Q) × F q (Q) satisfying a π (ñ) = a π ′ (ñ) forñ ∈ I S lies along the diagonal π = π ′ . It is shown in [46] that an admissible value for the exponent B is 2n + ε, for any ε > 0. In fact, their result can be refined, under the assumption that π f and π ′ f have the same (finite) conductor q and central character χ. Indeed, in this case, the bounds of Theorem B.1 of Appendix B save Norm(q) 2 off of this.
A.4.2. Second step. As in §A.1, we let u π = v π v π , v π 1/2 be the projection of the vector v π to the unit sphere in V . We now proceed to show that the vectors u π and u π ′ (for π = π ′ ) are quasi-orthogonal, in a quantifiable sense.
From our recurrence hypothesis, we deduce that K − 1 ≤ M − 1, as claimed.
Remark. The above induction argument works under the more general hypothesis that It is easy to see that the exponent of 2n in Theorem A.1 can be improved to n + 1 under these assumptions, and that the discriminant dependence is as described there. This is due to the fact that, under Riemann and Ramanujan, the map π → v π is injective as soon as X ≫ log 2 Q (see, for example, [32,Proposition 5.22]). This replaces step (1) in the proof of Theorem A.1. On the other hand, the proof of Proposition A.4 is insensitive to the Riemann hypothesis and the Ramanujan conjecture, despite the fact that the residue of the L(s, π ×π) is bounded below by 1/ log Q under these assumptions (see [32,Theorem 5.19]). In any case, with Theorem A.3 improved, we may take X = (D −n 2 Norm(q) −2 Q 2n ) 1/2+ε in executing step (3). Indeed, the exponent of Q required for the value of X in step (3)  Let F be a locally compact non-Archimedean field, and n, m two positive integers. Let π be a smooth irreducible representation of GL n (F ), with central character ω π and Artin conductor Ar(π) = a, and let ρ be a smooth irreducible representation of GL m (F ), with central character ω ρ and Artin conductor Ar(ρ) = b.
That bound cannot be improved in general but here, prompted by a query of F. Brumley, we improve (B.1) under an additional hypothesis.
When n = m and a = b, this gives Ar(π × ρ) ≤ (2n − 2)a, as used in the main text. Note also that when n = m = 1 the hypothesis implies a = b and Ar(π × ρ) = 0, which is fortunate since the right hand side of (B.2) is also 0! Thanks to the Langlands correspondence, we may express the theorem in terms of Weil-Deligne representations, and we indeed use that language in the proofs. We fix a separable algebraic closure F sep of F and let W F be the Weil group of F sep over F . We write σ, τ for the Weil-Deligne representations corresponding to π, ρ: they are directs sums of indecomposable Weil-Deligne representations. The theorem above is then equivalent to Theorem B.2. Assume that det σ det τ is unramified. Then Ar(σ ⊗ τ ) ≤ ma + nb − 2 min(a, b).
Remark. Assume that σ is the direct sum of characters of W F , all trivial but one, which then has to be det σ. Take for τ the contragredientσ of σ. Then Ar(σ ⊗σ) = (2n − 2)a, so one cannot improve (B.3) or (B.2) in general, even assuming that τ =σ.
We now proceed to the proof, relying on the results and techniques of [10]. Lemma B.4. Assume σ indecomposable. Then Ar(det σ) ≤ a/n.
Proof. By [10, Fact 2.1] and the notation there, we have σ = St r (σ ′ ), for some positive integer r and some irreducible representation σ ′ of W F . If σ ′ is an unramified character of W F then r = n and a = n − 1, whereas det σ is unramified, so Ar(det σ) = 0 ≤ a n . If σ ′ is not an unramified character, then a = rAr(σ ′ ) and det σ = (det σ) r , so it is enough to treat the case where σ = σ ′ is irreducible (and not unramified). But then a − n is the Swan exponent of σ, so, using [10, Fact 2.3], a n − 1 = inf{ε > 0 : σ(W ε F ) = 1}.
Since det σ is certainly trivial on the ramification subgroup W ε F if σ is, we see that the Swan exponent of det σ is at most a n − 1, so Ar(det σ) ≤ a/n.
Let us define the list of slopes σ. When indecomposable, σ has a list of n slopes, all equal to a/n. In general the list of slopes of σ is obtained by gathering the lists of slopes of its indecomposable summands, in increasing order. We write (a 1 , . . . , a n ) for the list of slopes of σ, and (b 1 , . . . , b m ) for the list of slopes of τ ; in particular, a = a 1 + · · · + a n and b = b 1 + · · · + b m .
B.2. In this n o , we assume n = 1. As the case n = m = 1 is done, we also assume m > 1. We first deal with the case b m−1 < b m . Then we can write τ = τ ′ ⊕ η for a character η of W F with Ar(η) = b m . By the Corollary B.5 (i), Ar(det τ ′ ) ≤ b m−1 and since det τ = (det τ ′ )η we get Ar(det τ ) = b m . But σ = det σ and det σ det τ is unramified, so we have a = b m .
On B.3. From now on we assume n > 1 and m > 1.
Indeed, since m−d i=1 max(a n , b i ) ≤ (m − d)a n + b ′ and d ≥ 2, we have 2 min(a ′ + a n , b ′ + db m ) + m−d i=1 max(a n , b i ) ≤ 2(a ′ + a n ) + (m − d)a n + b ′ ≤ ma n + da ′ + b ′ , establishing (B.6). By symmetry, the case where a n−1 = a n but b m−1 < b m also holds.
B.5. The final case is when a n−1 = a n and b m−1 = b m . Here the hypothesis that det σ det τ is unramified plays no role. By symmetry we may and do assume a n ≤ b m . We write σ = σ ′ ⊕ χ for a Weil-Deligne representation χ with dimension e ≥ 2 and all slopes equal to a n , and τ = τ ′ ⊕ η as in §B.4. We put a ′ = Ar(σ ′ ), b ′ = Ar(τ ′ ), so that a = a ′ + ea n , b = b ′ + db m .
We claim that the right-hand side of (B.7) is at most ma ′ + mea n + nb ′ + ndb m − min(a ′ + ea n , b ′ + db m ). This is equivalent to the inequality (B.8) da ′ + mea n + eb ′ + min(a ′ , b ′ ) ≥ min(a ′ + ea n , b ′ + db m ) + e m−d i=1 max(a n , b i ).
B.6. With an entirely similar reasoning, but replacing Artin exponents Ar with Swan exponents Sw, we get the following result, improving [10, Theorem CS] in a special case.