TY - JOUR

T1 - Special values of motivic L-functions and zeta-polynomials for symmetric powers of elliptic curves

AU - Löbrich, Steffen

AU - Ma, Wenjun

AU - Thorner, Jesse

N1 - Funding Information:
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. was supported by the Fulbright Commission and the Deutsche Forschungsgemeinschaft (DFG) Grant No. BR 4082/3-1. He thanks the Department of Mathematics and Computer Science at Emory University for their hospitality. W.M. thanks the Chinese Scholarship Council for its generous support and the Department of Mathematics and Computer Science at Emory University for their hospitality. J.T. is supported by a NSF Mathematical Sciences Postdoctoral Research Fellowship.
Publisher Copyright:
© 2017, The Author(s).

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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(AQ). 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 R(s) = 1 / 2.

AB - 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(AQ). 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 R(s) = 1 / 2.

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U2 - 10.1186/s40687-017-0114-0

DO - 10.1186/s40687-017-0114-0

M3 - Article

AN - SCOPUS:85050392497

SN - 2522-0144

VL - 4

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

IS - 1

M1 - 26

ER -