Special values of motivic L-functions and zeta-polynomials for symmetric powers of elliptic curves

Abstract

Let M be a pure motive over Q of odd weight w ≥ 3, even rank d ≥ 2, and global conductor N whose L-function L (s, M) coincides with the L-function of a self-dual algebraic tempered cuspidal symplectic representation of GL d (A Q ). We show that a certain polynomial which generates special values of L(s, 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 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 → 1 − s, and all of their zeros on the line (s) = 1/2.

Asymptotics for the Dirichlet coefficients of symmetric power $L$-functions

“…The assertions for j = 1, 2, 3, 4, and (2.19) for 1 ≤ j ≤ 8, are proved in Proposition 2.1 of Lau and Wu [26]. The remaining parts are easily deduced from [3,4,7,27] and the properties RS 2 and RS 3 of Rudnick-Sarnak [30].…”

“…It is generally conjectured that the function L(sym j f, s) is an entire function and satisfies a functional equation, which is a special case of Langlands functoriality and is proved for j = 1, 2, 3, 4 (see [14,10,18,19,17]). By using these properties of symmetric power L-functions and the Landau theorem, Recently there have been some breakthroughs on the automorphy of sym j f by Dieulefait [7], Clozel and Thorne [3,4], Löbrich, Ma and Thorner [27]; more precisely, for j = 5, 6, 7, 8, sym j f is an automorphic cuspidal representation of GL(j + 1). This implies that the jth symmetric power L-function and its Rankin-Selberg L-function have meromorphic continuations to the whole complex plane, and satisfy certain functional equations.…”

“…For j = 1, 2, 3, 4, Lemma 2.3 follows from Iwaniec [14] for j = 1, Gelbart and Jacquet [10] for j = 2 and Kim and Shahidi [17,18,19] for j = 3, 4, respectively. According to Clozel and Thorne [3,4], Dieulefait [7], Löbrich, Ma and Thorner [27], for j = 5, 6, 7, 8, sym j f is an automorphic cuspidal representation of GL(j + 1). This implies that the jth symmetric power L-function is a holomorphic function and satisfies a functional equation.…”