Zetapolynomials for modular form periods
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 zetapolynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1s)$, 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 zetafunctions. 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 BlochKato Tamagawa Number Conjecture, they encode the arithmetic of a combinatorial arithmeticgeometric object which we call the "BlochKato complex" for $f$. Loosely speaking, these are graded sums of weighted moments of orders of ŠafarevičTate groups associated to the Tate twists of the modular motives.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.00752
 arXiv:
 arXiv:1602.00752
 Bibcode:
 2016arXiv160200752O
 Keywords:

 Mathematics  Number Theory;
 11F11;
 11F67
 EPrint:
 15 pages, 3 figures. Minor edits in v2, to appear in Advances in Mathematics. arXiv admin note: text overlap with arXiv:1605.05536