For more information about this meeting, contact Robert Vaughan, Mihran Papikian, Ae Ja Yee, Kirsten Eisentraeger.

Title: | Zeta-polynomials for modular form periods |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Larry G. Rolen III, Penn State University |

Abstract: |

Answering problems of Manin, we use the critical $L$-values of even weight newforms $f$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(\rho)=0$, then $\Re(\rho)=1/2$. 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 zeta-functions.
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 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 \v{S}afarevi\v{c}--Tate groups associated to the Tate twists of the modular motives. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 02 / 25 / 2016 |

Time: | 11:15am - 12:05pm |