# Number Theory Seminar

## Spring 2023

Date: | March 2, 2023 |

Time: | 2:30pm |

Location: | BLOC 302 |

Speaker: | Larry Rolen, Vanderbilt University |

Title: | Recent problems in partitions and other combinatorial functions |

Abstract: | In this talk, I will discuss recent work, joint with a number of collaborators, on analytic and combinatorial properties of the partition and related functions. This includes work on recent conjectures of Stanton, which aim to give a deeper understanding into the "rank" and "crank" functions which "explain" the famous partition congruences of Ramanujan. I will describe progress in producing such functions for other combinatorial functions using the theory of modular and Jacobi forms and recent connections with Lie-theoretic objects due to Gritsenko-Skoruppa-Zagier. I will also discuss how analytic questions about partitions can be used to study Stanton's conjectures, as well as recent conjectures on partition inequalities due to Chern-Fu-Tang and Heim-Neuhauser, which are related to the Nekrasov-Okounkov formula. |

Date: | March 9, 2023 |

Time: | 2:30pm |

Location: | BLOC 302 |

Speaker: | Yen-Tsung Chen, Penn State University |

Title: | On almost holomorphic Drinfeld modular forms and their special values at CM points |

Abstract: | It is known that the derivative of a modular form is no longer modular. One possible approach to this issue is modifying the differential operator by sacrificing the holomorphy. This leads to the study of nearly holomorphic modular forms and Maass-Shimura operators. The former object was introduced and investigated by Shimura in the 1970s. It also has a strong connection with the quasi-modular forms introduced by Kaneko and Zagier in 1995. In this talk, we introduce the notion of almost holomorphic Drinfeld modular forms and construct an analogue of the Maass-Shimura operators in this context. The structure of almost holomorphic Drinfeld modular forms as well as its connection with Drinfeld quasi-modular forms have been developed. Furthermore, we establish the algebraic independence result for the special values of almost holomorphic Drinfeld modular forms at CM points. This is joint work with Oguz Gezmis. |

Date: | April 13, 2023 |

Time: | 2:30pm |

Location: | BLOC 302 |

Speaker: | Matthew Kroesche, Texas A&M University |

Title: | Low-lying zeros of a family of automorphic forms |

Abstract: | In their paper “Low-Lying Zeros of Families of L-Functions”, Iwaniec, Luo, and Sarnak calculated the distribution of low-lying zeros of families of modular L-functions in level aspect for test functions with support contained in (−2, 2). In this work, we extend this result to thin (in the p-adic sense) subfamilies of these families, obtained by twisting lower-level modular forms by a character, and investigate what can be said about the support and the lower-order terms in the one-level density in this case. |

Date: | April 20, 2023 |

Time: | 2:30pm |

Location: | BLOC 302 |

Speaker: | Wei-Cheng Huang, Texas A&M University |

Title: | Regulators of tensor/symmetric/alternating squares of Drinfeld modules of rank 2 |

Abstract: | Regulators of Drinfeld modules or t-modules appear in the class module formulas of Demeslay, Fang, and Taelman. In this talk, we will investigate regulators of the tensor, symmetric, and alternating squares of Drinfeld modules of rank 2, and express them in terms of data from the Drinfeld modules themselves. |

Date: | April 27, 2023 |

Time: | 2:30pm |

Location: | BLOC 302 |

Speaker: | Agniva Dasgupta, Texas A&M University |

Title: | Second Moment of Twisted Cusp Forms Along a Coset |

Abstract: | We prove Lindelöf-on-average upper bound for the second moment of the L-function associated to a level 1 holomorphic cusp form, twisted along a coset of subgroup of the characters modulo q^{2/3} (where q = p^3 for some odd prime p). This result should be seen as a q-aspect analogue of Anton Good’s (1982) result on upper bounds of the second moment of cusp forms in short intervals. |

Date: | May 4, 2023 |

Time: | 2:30pm |

Location: | BLOC 302 |

Speaker: | Matias Alvarado, Pontificia Universidad Católica de Chile |

Title: | Zsigmondy bounds for families of Drinfeld modules of rank 2 |

Abstract: | The Zsigmondy sets are objects of study in arithmetic dynamics. Ji & Zhao proved that these sets associated with Drinfeld modules are finite, but unfortunately, the proof is not effective. In our work, we give a bound for the Zsigmondy set in the case of certain families of Drinfeld modules of rank 2. In this talk, we will define the Zsigmondy set in a dynamic, visit the results of Ji-Zhao, and review some preliminaries to understand the strategy to approach the problem. At the end of the talk, we will sketch the proof of our theorem. |