Zeta-polynomials for modular form periods

Ken Ono, Larry Rolen, Florian Sprung

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Answering problems of Manin, we use the critical L-values of even weight k≥4 newforms f∈Sk0(N)) to define zeta-polynomials Zf(s) which satisfy the functional equation Zf(s)=±Zf(1−s), and which obey the Riemann Hypothesis: if Zf(ρ)=0, then Re(ρ)=1/2. The zeros of the 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) 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.

Original languageEnglish (US)
Pages (from-to)328-343
Number of pages16
JournalAdvances in Mathematics
StatePublished - Jan 14 2017

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Bloch–Kato complex
  • Ehrhart polynomials
  • Modular forms
  • Period polynomials
  • Zeta-polynomials


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