Zeta-polynomials for modular form periods

Ken Ono, Larry Rolen, Florian Sprung

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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
Volume306
DOIs
StatePublished - Jan 14 2017
Externally publishedYes

Fingerprint

Modular Forms
Signed number
Ehrhart Polynomial
Moment
Stirling numbers
Integer Points
Polynomial
Riemann hypothesis
Polytopes
Twist
Riemann zeta function
Functional equation
Analogy
Line
Zero
Object

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Zeta-polynomials for modular form periods. / Ono, Ken; Rolen, Larry; Sprung, Florian.

In: Advances in Mathematics, Vol. 306, 14.01.2017, p. 328-343.

Research output: Contribution to journalArticle

Ono, Ken ; Rolen, Larry ; Sprung, Florian. / Zeta-polynomials for modular form periods. In: Advances in Mathematics. 2017 ; Vol. 306. pp. 328-343.
@article{4e3d33f345fa4197885b6029c92ff5f1,
title = "Zeta-polynomials for modular form periods",
abstract = "Answering problems of Manin, we use the critical L-values of even weight k≥4 newforms f∈Sk(Γ0(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.",
keywords = "Bloch–Kato complex, Ehrhart polynomials, Modular forms, Period polynomials, Zeta-polynomials",
author = "Ken Ono and Larry Rolen and Florian Sprung",
year = "2017",
month = "1",
day = "14",
doi = "10.1016/j.aim.2016.10.004",
language = "English (US)",
volume = "306",
pages = "328--343",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Zeta-polynomials for modular form periods

AU - Ono, Ken

AU - Rolen, Larry

AU - Sprung, Florian

PY - 2017/1/14

Y1 - 2017/1/14

N2 - Answering problems of Manin, we use the critical L-values of even weight k≥4 newforms f∈Sk(Γ0(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.

AB - Answering problems of Manin, we use the critical L-values of even weight k≥4 newforms f∈Sk(Γ0(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.

KW - Bloch–Kato complex

KW - Ehrhart polynomials

KW - Modular forms

KW - Period polynomials

KW - Zeta-polynomials

UR - http://www.scopus.com/inward/record.url?scp=84994226752&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994226752&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.10.004

DO - 10.1016/j.aim.2016.10.004

M3 - Article

VL - 306

SP - 328

EP - 343

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -