Zeta-polynomials for modular form periods

TL;DR

This paper introduces zeta-polynomials derived from the critical L-values of modular forms, which satisfy a functional equation and the Riemann Hypothesis, revealing deep arithmetic information related to the Bloch-Kato conjecture.

Contribution

It constructs and analyzes zeta-polynomials for modular form periods that encode arithmetic data and satisfy key properties like the functional equation and Riemann Hypothesis.

Findings

Zeta-polynomials satisfy the functional equation and RH.

Zeros are distributed similarly to classical zeta-functions.

Polynomials encode arithmetic information related to Bloch-Kato conjecture.

Abstract

Answering problems of Manin, we use the critical $L$-values of even weight $k\geq 4$ newforms $f\in S_k(\Gamma_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(\rho)=0$, then $\operatorname{Re}(\rho)=1/2$. The zeros of the $Z_f(s)$ on the critical line in $t$-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and "weighted moments" of critical values $L$-values. In analogy with Ehrhart polynomials which keep track of integer points in polytopes, the $Z_f(s)$ keep track of arithmetic information. Assuming the Bloch--Kato Tamagawa Number Conjecture, they encode the arithmetic of a combinatorial arithmetic-geometric object which we call the "Bloch-Kato complex" for $f$. Loosely speaking, these are graded sums of weighted moments of orders of \v{S}afarevi\v{c}-Tate groups associated to the Tate twists of the modular motives.