Zeta-polynomials for modular form periods

详细信息    查看全文

• 作者：Ken Ono^a ; ^{ono@mathcs.emory.edu} ; Larry Rolen^b ; ^{larryrolen@psu.edu} ; Florian Sprung^c ; ^{fsprung@princeton.edu}
• 关键词：11F11 ; 11F67
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：328-343
• 全文大小：519 K
• 卷排序：306

文摘

Answering problems of Manin, we use the critical L  -values of even weight k≥4k≥4 newforms f∈Sk(Γ0(N))f∈Sk(Γ0(N)) to define zeta-polynomials Zf(s)Zf(s) which satisfy the functional equation Zf(s)=±Zf(1−s)Zf(s)=±Zf(1−s), and which obey the Riemann Hypothesis: if Zf(ρ)=0Zf(ρ)=0, then Re(ρ)=1/2Re(ρ)=1/2. The zeros of the Zf(s)Zf(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 L  -values. In analogy with Ehrhart polynomials which keep track of integer points in polytopes, the Zf(s)Zf(s) encode 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 Šafarevič–Tate groups associated to the Tate twists of the modular motives.