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

Steffen Löbrich, Wenjun Ma, Jesse Thorner

Research output: Contribution to journalArticlepeer-review

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(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.

Original languageEnglish (US)
Article number26
JournalResearch in Mathematical Sciences
Volume4
Issue number1
DOIs
StatePublished - Dec 1 2017
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (miscellaneous)
  • Computational Mathematics
  • Applied Mathematics

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