### Abstract

Answering problems of Manin, we use the critical L-values of even weight k≥4 newforms f∈S_{k}(Γ_{0}(N)) to define zeta-polynomials Z_{f}(s) which satisfy the functional equation Z_{f}(s)=±Z_{f}(1−s), and which obey the Riemann Hypothesis: if Z_{f}(ρ)=0, then Re(ρ)=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 L-values. In analogy with Ehrhart polynomials which keep track of integer points in polytopes, the Z_{f}(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 language | English (US) |
---|---|

Pages (from-to) | 328-343 |

Number of pages | 16 |

Journal | Advances in Mathematics |

Volume | 306 |

DOIs | |

State | Published - Jan 14 2017 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*306*, 328-343. https://doi.org/10.1016/j.aim.2016.10.004

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

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 306, pp. 328-343. https://doi.org/10.1016/j.aim.2016.10.004

}

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 -