Special Values of Motivic $L$-Functions and Zeta-Polynomials for Symmetric Powers of Elliptic Curves

Let $\mathcal{M}$ be a pure motive over $\mathbb{Q}$ of odd weight $w\geq 3$, even rank $d\geq 2$, and global conductor $N$ whose $L$-function $L(s,\mathcal{M})$ coincides with the $L$-function of a self-dual algebraic tempered cuspidal symplectic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$. We show that a certain polynomial which generates special values of $L(s,\mathcal{M})$ (including all of the critical values) has all of its zeros equidistributed on the unit circle, provided that $N$ or $w$ are sufficiently large with respect to $d$. These special values have arithmetic significance in the context of the Bloch-Kato conjecture. We focus on applications to symmetric powers of semistable elliptic curves over $\mathbb{Q}$. Using the Rodriguez-Villegas transform, we use these results to construct large classes of"zeta-polynomials"(in the sense of Manin) arising from symmetric powers of semistable elliptic curves; these polynomials have a functional equation relating $s\mapsto 1-s$, and all of their zeros on the line $\operatorname{Re}(s)=1/2$.


Introduction and statement of results
Let f (z) = ∞ n=1 a f (n)q n be a normalized holomorphic cuspidal modular form of even weight k ≥ 2 and level N, and trivial nebentypus. Assume further that f is an eigenform for the Hecke operators T p for p ∤ N and U p for all p | N. We call such a modular form a newform. The L-function L(s, f ) associated to a newform f , which is given by f (τ )(τ − z) k−2 dτ , which is a polynomial of degree k − 2 in z. Using (1.3), we expand (τ − z) k−2 to obtain (1.4) r By expressing Λ(s, f ) in terms of L(s, f ) via (1.2), we see that r f (z) is a generating function for the critical values L(1, f ), L(2, f ), . . . , L(k −1, f ). For additional background and details, see [12] and the sources contained therein.
It follows from the functional equation for Λ(s, f ) that r f (z) satisfies a functional equation of its own, relating r f ( z i √ N ) to r f ( 1 iz √ N ) and fixing the unit circle S 1 = {z ∈ C : |z| = 1}. In analogy with the expected behavior of the nontrivial zeros of the Riemann zeta function ζ(s) or the nontrivial zeros of L(s, f ), one might expect that all of the zeros of r f ( z i √ N ) lie on S 1 . Because of the similarity with the Riemann hypothesis, this has been called the Riemann hypothesis for period polynomials. Conrey, Farmer, and Imamoglu [7] proved result of this sort for the odd part of r f ( z i √ N ), and the Riemann hypothesis for the period polynomials associated to newforms of level 1 and even weight k ≥ 2 was established by El-Guindy and Raji [10]. The Riemann hypothesis for period polynomials is now a theorem due to Jin, Ma, Ono, and Soundararajan [12] for all newforms of weight k ≥ 2 with trivial nebentypus; furthermore, they proved that if either k or N is sufficiently large, then the zeros of r f ( z i √ N ) are equidistributed on S 1 .
The truth of the Riemann hypothesis for period polynomials, along with the statement of equidistribution, introduces strong conditions on the sizes of the critical values L(1, f ), L(2, f ), . . . , L(k − 1, f ); these values have significance in algebraic number theory and arithmetic geometry. For newforms f of weight 2 associated to elliptic curves, r f (z) is a constant polynomial with a non-zero factor of L(1, f ). If the Birch and Swinnerton-Dyer conjecture is true, then L(1, f ) encapsulates much of the arithmetic of the elliptic curve, including order of the Tate-Shafarevich group and whether or not the rank of the Mordell-Weil group is positive. Unfortunately, the results in [12] cannot provide insight into the Birch and Swinnerton-Dyer conjecture, because for k = 2, the period polynomial is constant. Thus the Riemann hypothesis for period polynomials when k = 2 is trivially satisfied without shedding light on L(1, f ). If k ≥ 4, the critical values hold similar importance in the context of the Bloch-Kato conjecture [3], which generalizes of the Birch and Swinnerton-Dyer conjecture.
In this paper, we use the ideas in [12] to study critical values of motivic L-functions. It is well-known that each modular L-function L(s, f ) is attached to a certain pure motive over Q of weight k − 1, conductor N, and rank 2; furthermore, L(s, f ) is the L-function of a certain cuspidal automorphic representation of GL 2 (A Q ). (Here, A Q denotes the ring of adeles of Q.) The critical values of motivic L-functions carry similar arithmetic significance in the context of the Bloch-Kato conjecture. When motivic L-functions coincide with automorphic L-functions, they have important analytic properties which generalize those of L(s, f ). However, there does not appear to be a canonical generating polynomial for critical values of motivic L-functions that generalizes the properties of r f (z). Thus we construct a polynomial p M (z) (see (3.1)) which mimics r f ( z i √ N ) and prove the following. Theorem 1.1. Let M be a pure motive over Q of odd motivic weight w = 2m + 1 ≥ 3, even rank d ≥ 2, global conductor N, and Hodge numbers h ν for 0 ≤ ν ≤ m (see Section 2). Suppose that the L-function L(s, M) of M coincides with the L-function of an algebraic, tempered, cuspidal symplectic representation of GL d (A Q ). Let p M (z) be the polynomial defined in (3.1).
(1) If m = 1 and h 0 ∈ {0, 1}, then the zeros of p M (z) lie on S 1 and tend to be equidistributed as N → ∞.
(2) If m ≥ 2, 2m hm ≥ (1 + 1/m) h 0 , and N > A d m (where A m is defined by (4.4)), then the zeros of p M (z) lie on S 1 and tend to be equidistributed as N → ∞. Remark. If L(s, M) is the L-function of a newform of (modular) weight k ≥ 4, then p M (z) reduces to a constant multiple of r f ( z i √ N ), whose zeros are studied in [12]. It is unclear how to ensure that all of the zeros lie on S 1 while maintaining uniformity in d when m ≥ 2 and d is large compared to log N. Despite this setback, we already have a result that is strong enough to address a natural family of examples, namely the odd symmetric power L-functions L(s, Sym n f ) of the newforms f considered in [12] that do not have complex multiplication (CM). The next result follows from Theorem 1.1 in case of M = Sym n f and n odd. Corollary 1.2. Let n ≥ 3 be an odd integer and f a non-CM newform of even integral weight k ≥ 2, squarefree level N ≥ 13, trivial nebentypus, and integral Fourier coefficients. We assume that N ≥ 46 if (k, n) = (2,5) and N ≥ 17 if (k, n) ∈ {(2, 7), (4, 3)}. If L(s, Sym n f ) is the L-function of an algebraic tempered cuspidal symplectic representation of GL n+1 (A Q ), then all of the zeros of p Sym n f (z) lie on S 1 . The zeros tend to be equidistributed as n or N goes to ∞.
We find the most interesting case to be when k = 2 because the results in [12] are trivial in this case. By numerically checking the cases that are not covered by Corollary 1.2, we obtain the following result. Theorem 1.3. Let E/Q be a non-CM elliptic curve of squarefree conductor N, and let n ≥ 3 be an odd integer. If L(s, Sym n E) is the L-function of an algebraic, tempered, cuspidal symplectic representation of GL n+1 (A Q ), then all of the zeros of p Sym n E (z) given by (3.1) lie on S 1 . The zeros tend to be equidistributed as n or N goes to ∞.
In [15], Manin speculated on the existence of zeta-polynomials Z(s) which (in analogy with expected behavior of the Riemann zeta function and L(s, f )) satisfy a functional equation of the form Z(s) = ±Z(1 − s) and have all of their zeros lie on the line ℜ(s) = 1/2. Furthermore, there should be a "nice" generating function for the sequence {Z(−n)} ∞ n=1 along with an arithmetic-geometric interpretation of Z(−n). Manin constructed zeta-polynomials by applying the "Rodriguez-Villegas transform" [20] to the odd part of the period polynomial of a newform using the results in [7]; he suggests that these polynomials arise from non-Tate motives and geometric objects lying below Spec Z but not over F 1 .
Manin asked whether there exist zeta-polynomials which can be canonically constructed from the full period polynomial. Ono, Rolen, and Sprung [18] recently used the results in [12] to address this question, producing a large class of zeta-polynomials canonically constructed from the critical values of classical newforms f . Assuming the Bloch-Kato conjecture, these zeta-polynomials encode further Galois cohomological structure of Selmer groups for Tate-twists that have been assembled as Stirling complexes. Moreover, in analogy with the Maclaurin expansion the zeta-polynomials Z f (s) constructed in [18] satisfy Using Theorem 1.3, we construct zeta-polynomials arising from the special values of odd symmetric power L-functions of semistable elliptic curves over Q. Using the Bloch-Kato conjecture, one can express the coefficients of these zeta-polynomials in terms of Tamagawa numbers and generalized Shafarevich-Tate groups of the symmetric powers. Theorem 1.4. Let E/Q be a non-CM elliptic curve, and let n ≥ 3 be odd. Suppose that L(s, Sym n E) is the L-function of an algebraic, tempered, cuspidal symplectic representation of GL n+1 (A Q ). Let Z Sym n E (s) be the polynomial defined by (7.1). The following are true.
(1) For all s ∈ C, we have that is the sign of the functional equation for L(s, Sym n E).
We review motivic L-functions and their conjectured analytic properties in Section 2. In Section 3, we prove some lemmas that are needed for the proofs of Theorem 1.1, which we prove in Sections 4 and 5. We then discuss symmetric power L-functions and prove Theorems 1.3 and 1.4 in Sections 6 and 7.
Acknowledgements. This project was suggested by Ken Ono; the authors thank him for his comments and support. The authors also thank Seokho Jin, Robert Lemke Oliver, Akshay Venkatesh, and Don Zagier for helpful conversations. S.L. thanks the Fulbright Commission for their generous support and the Department of Mathematics and Computer Science at Emory University for their hospitality. W.M. thanks the Chinese Scholarship Council for its generous support. J.T. is supported by a NSF Mathematical Sciences Postdoctoral Research Fellowship.

Motivic L-functions
We begin by recalling the conjectural properties of motivic L-functions. For more details, see Serre [22] and Iwaniec and Kowalski [11, Chapter 5].
2.1. Conjectured analytic properties. Define a pure motive M over Q of weight w, rank d, and global conductor N by specifying Betti, de Rham, and ℓ-adic realizations (for each prime ℓ) which are vector spaces of dimension d over Q, Q, and Q ℓ , respectively; each is endowed with additional structures and comparison isomorphisms as in [5,8]. In particular, H B (M) admits an involution ρ B , H ℓ (M) is a Gal(Q/Q)-module, and there is a Hodge decomposition into C-vector spaces The involution ρ B acts on H i,j (M) by ρ B (H i,j (M)) = H j,i (M). When w is even, this tells us that H w/2,w/2 (M) is invariant under ρ B ; when w is odd, we take H w/2,w/2 (M) = {0}. If w is even and H w/2,w/2 (M) = {0}, then the involution ρ B acts on H w/2,w/2 (M) by α ∈ {−1, 1}; we then define the quantity b ± (M) by We denote by ρ ℓ the representation which induces the Gal(Q/Q)-module structure on H ℓ (M). For any prime p, let Frob p ∈ Gal(Q/Q) be the Frobenius element at p, which is defined modulo conjugation and modulo the inertia subgroup One typically assumes (and expects) that Because we consider M over Q, the degree of L(s, M) also equals We now describe the hypotheses for L(s, M) which are crucial to our arguments. When w is odd, we see that d must be even (see (2.1)). Now, consider now the exterior square representation Ext 2 (π M ) and the Euler product where at each prime p ∤ N we have We know that L(s, Ext 2 (π M )) has a meromorphic continuation to C with no poles outside of the set { w ′ 2 , w ′ 2 + 1}, where w ′ is the weight of Ext 2 (π M ) [17]. If L(s, Ext 2 (π M )) has a pole at s = w 2 + 1, then π M is a cuspidal symplectic representation of GL d (A Q ); let A s d (Q) denote the set of such representations. For any π M ∈ A s d (Q), Lapid and Rallis [14] proved that Λ( w+1 2 , π M ) ≥ 0. (This vastly generalizes a result of Waldsuprger [24] for L-functions of newforms.) Therefore, the hypotheses of Theorem 1.1 succinctly describe the most natural class of motivic L-functions for which the methods in [12] can be used for studying special and critical values.
In Theorem 1.1, we require that 2m hm ≥ (1 + 1/m) h 0 . This not true of all M. In fact, for any integer m ≥ 0 and any collection of nonnegative integers h 0 , . . . , h m , there exists a motive of weight 2m + 1 with Hodge numbers h 0 , . . . , h m ; see Arapura [1] and Schreieder [21] for explicit constructions. However, for newforms and their symmetric powers (see Section 6) as well as many other interesting cases, we have h ν ∈ {0, 1} for each 1 ≤ ν ≤ m.

Preliminary Lemmas and Setup
Let M be a pure motive over Q of rank d ≥ 2 with global conductor N, odd weight w = 2m + 1 ≥ 3, root number ε = ε(M), and Hodge numbers h ν for 0 ≤ ν ≤ m. (It will be more notationally convenient for us to use m instead of w.) For convenience, we let S 1 := {z ∈ C : |z| = 1} and D := {z ∈ C : |z| < 1}.
We now define our first analogue of (1.4) by letting where ζ(s) is the Riemann zeta function.
Proof. The Euler product for L(s, M) gives rise to the function Λ M (n) which is defined by the Dirichlet series identity One sees that |Λ M (n)| ≤ dn w/2 Λ(n) for all n ≥ 1, where Λ(n) is the usual von Mangoldt function; this estimate follows from Part (4) of Hypothesis 2.1. Let 0 < a ≤ t ≤ b. By the above discussion, Consequently, which equals the right hand side of the desired inequality.

Proof of Theorem 1.1 when N is large
Our proof of Theorem 1.1 is broken into two cases. First we consider the case when m = 1, in which case P M (z) is linear. Then we consider the case where m ≥ 2.

By Lemma 3.2 we have
Since Λ(m + 1, M) ≤ Λ(m + 2, M), (4.3) is satisfied when 2m hm ≥ (1 + 1/m) h 0 , as can be seen using term-by-term comparison. This completes the proof. It is straightforward to compute A 2 ≤ 23.83, A 3 ≤ 11.92, A m ≤ 8 for m ≥ 4, and lim m→∞ A m = 2π. Thus the above proof cannot produce a lower bound for N better than (2π) d ; we must handle the cases where N ≤ A d m differently.

Proof of Theorem 1.1 when m is large
On the unit circle, r f (z) is well-approximated by an exponential function [12, Section 6], but if M is arbitrary, then p M (z) is well-approximated on the unit circle by a certain generalized hypergeometric function. Unfortunately, it is computationally intractable to locate the zeros of the real and imaginary parts of generalized hypergeometric functions, and Rouché's Theorem only gives us the zeros of the real and imaginary part simultaneously. Therefore, we can only prove that "most" zeros (depending on d and N) lie on the unit circle as the weight becomes large.
Let d be fixed. If we define (5.1) then we may write P M (z) as which we approximate by its partial sums T m,d,N (z) := m It follows from [10, Theorem 2.2] that p M (z) has as many zeros on S 1 as Q M (z) has inside D. Thus Part 3 of Theorem 1.1 follows from the following statement. Proof. We use Rouché's Theorem. First, for |z| = 1, we estimate with Lemma 3.2 The function x → 2 x (ζ( 1 2 + x) d − 1) is monotonically decreasing for x ≥ 1, so Furthermore, If d is fixed, then both (5.5) and (5.6) can be made arbitrarily small if m is sufficiently large.
We first assume that F d,N has no zeros on S 1 . Since T m,d,N (z) converges to F d,N (z) locally uniformly as m tends to infinity, we have If F d,N has zeros on S 1 , then we choose an r > 1, such that all the zeros of F d,N in the region {r −1 ≤ |z| ≤ r} lie on S 1 and slightly modify the argument above by applying Rouché's Theorem to the circle {|z| = r}.
By taking d = 2, we have that F 2,N (z) = exp( 2π √ N z). Since F 2,N (z) has no zeros in D, we have that c d,N = 0; thus p M (z) has all of its zeros on S 1 , as shown in [12]. However, for d = 4, the situation already becomes noticeably more complicated; when d = 4, we have that F 4,N (z) = I 0 (4πN −1/4 √ z), where I 0 denotes the I-Bessel function. When d ≥ 6, F d,N (z) is a generalized hypergeometric function. To illustrate the difficulty when d ≥ 4, we directly compute To see how these compare with those of the previous section, we observe that 746 ≈ 1 2 (2π) 4 and 45607 ≈ 3 4 (2π) 6 . Thus it appears that the weight aspect of the results in [12] do not readily generalize to our setting when d is large.
6. Symmetric Power L-functions and the Proof of Theorem 1.3 6.1. Symmetric power L-functions of non-CM newforms. Let f be a non-CM newform of even weight k ≥ 2, squarefree level N, and trivial nebentypus. It is well-known that L(s, f ) is a motivic L-function satisfying Hypothesis 2.1 with weight w = k − 1, rank d = 2, and global conductor N. (See [12] and the sources contained therein).
For each prime ℓ, Deligne proved that there exists a representation ρ ℓ : Gal(Q/Q) → GL 2 (Z ℓ ) with the property that if p is a prime not dividing ℓN and Frob p is the Frobenius automorphism of Gal(Q/Q) at p, then the characteristic polynomial of ρ d (Frob p ) is x 2 − a f (p) + p k−1 . By Deligne's proof of the Weil Conjectures (which establishes Part 2 of Hypothesis 2.1), we know that |a f (p)| ≤ 2p (k−1)/2 . Thus the roots of the characteristic polynomial are α p p (k−1)/2 and β p p (k−1)/2 , where β p =ᾱ p and α p β p = 1. We recast the Euler product of L(s, f ) in (1.1) as When N is squarefree, the Euler product of the n-th symmetric power of f , which we denote by Sym n f , is given by Their critical values are important in the context of the Bloch-Kato conjecture, much like those of L(s, f ). (See [9] for an accessible overview along with some convincing computations.) The weight of Sym n f is n(k − 1), the rank is n + 1, and the global conductor is N n . (It is for this reason, and this reason alone, that we restrict N to be squarefree.) When n = 2r + 1 is odd, the integers which are critical for Sym 2r+1 f are r(k − 1) + j for 1 ≤ j ≤ k − 1. The Hodge numbers all lie in {0, 1}; see [6] for an exact expression for L ∞ (s, Sym n f ). From this we can check that the conditions of Theorem 1.1 (1) or (2) are satisfied under the assumptions of Corollary 1.2. Conjecturally, we have Sym n f ∈ A n+1 (Q) for each n ≥ 0, and Sym n f ∈ A s n+1 (Q) for each odd n ≥ 1. Unconditionally, we know that Sym n f ∈ A n+1 (Q) for each n ≤ 8 (see Clozel and Thorne [4], [6], and the sources contained therein). Moreover, as part of the celebrated proof of the Sato-Tate conjecture [2], we know that L(s, Sym n f ) can be analytically continued to the line ℜ(s) = 1 for each n ≥ 1. It follows from the Euler product representation of L(s, Sym n f ) and (2.3) that if n ≥ 1 is odd, then L(s, Ext 2 (Sym n f )) = ζ(s) In particular, if n is odd and Sym n f ∈ A n+1 (Q), then L(s, Ext 2 (Sym n f )) has a pole at s = 1. Thus by Lapid and Rallis [14], we have that Λ( n(k−1)+1 2 , f ) ≥ 0. For n = 1 and n = 3, these results were proved by Waldspurger [24] and Kim [13], respectively. Regardless of whether N is squarefree, we expect that L(s, Ext 2 (Sym n f )) has a pole at s = 1 for all odd n ≥ 1, in which case Sym n f ∈ A s n+1 (Q) and we obtain the desired nonvanishing at the central critical point.

6.2.
Proof of Theorem 1.3. By the modularity theorem, if E is a semistable elliptic curve of squarefree conductor N, then E corresponds to a weight 2 newform of level N, trivial nebentypus, and integral Fourier coefficients. Thus L(s, Sym n E) = L(s, Sym n f ). By Corollary 1.2, the only cases left to check are n = 5, 11 ≤ N ≤ 43 and n = 7, 11 ≤ N ≤ 15. We observe that in all of these exceptional cases except for (n, N) ∈ {(5, 37), (5, 43)}, corresponding to the isogeny classes 37.a and 43.a in Cremona's table, the root number ε(Sym n f ) is −1; these are stored on the L-function and Modular Form Database (LMFDB) website at http://www.lmfdb.org.
In the cases with ε(Sym n f ) = 1 (resp. n = 7), we explicitly compute the zeros of P Sym 5 f (resp. P Sym 7 f ) and observe that all of them lie in the open unit disc. For this, we use the critical value L(3, Sym 5 f ) and the Dirichlet coefficients of L(s, Sym 5 f ) (resp. L(s, Sym 7 f )), which are stored in the Lcalc files on http://www.lmfdb.org.
If n = 5 and ε(Sym 5 f ) = −1, we have  We now show that under the hypotheses of Theorem 1.3, p Sym n E (z) satisfies the hypotheses of Theorem 7.1.
Lemma 7.2. Let E/Q be a semistable elliptic curve, and suppose that Sym n E satisfies the hypotheses of Theorem 1.3. If ε(Sym n E) = 1, then p Sym n E (1) = 0. If ε(Sym n E) = −1, then p Sym n E (z) has a simple zero at z = 1.
Proof. Let n ≥ 3 be odd, let m = n−1 2 , and let ε = ε(Sym n E). By (3.2) and the fact that L(s, Sym n E) is self-dual, we have that p Sym n E (1) equals