# Honors Oral Exam

## On the discriminant of the Hecke Ring, $\mathbb{T}_{k}$, and its index in the Ring of Integers of $\mathbb{T}_{k} \otimes \mathbb{Q}$.

Gregory Michajlyszyn (University of Rochester)

Friday, May 8th, 2020
11:30 AM - 12:30 PM
https://rochester.zoom.us/j/98902034581

In the late 1930’s, Erich Hecke discovered a family of commutative operators on the spaces of modular forms, that provide an important bridge between the complex analysis and the underlying algebra inherent in the study of modular forms. When one fixes the weight and the level, these operators generate a commutative ring $\mathbb{T}_k$ which is isomorphic to a subring of finite index in the direct product of the rings of integers in a finite number of number fields.

All of the results in this talk stem from a single theorem of Tate and Serre (generalized by Jochnowitz to arbitrary levels) showing that even though there are an infinite number of systems of eigenvalues for the Hecke operators in characteristic 0, there are only a finite number mod $\ell$. This implies that even though the generalized eigenspaces for the Hecke operators in characteristic zero are all one dimensional, the dimensions of the generalized eigenspaces for varying weights when one works mod $\ell$ must be unbounded.

Jochnowitz was able to translate this into a theorem which showed that the power of a prime $\ell$ dividing the discriminant of the Hecke ring $\mathbb{T}_k$ of weight $k$ and arbitrary level grows linearly with $k$.

In this talk, after limiting ourselves for simplicity to the case of level 1, we present related results also due to Jochnowitz about the structure of the local components of the Artin rings $\mathbb{T}_k / \ell \mathbb{T}_k$ or equivalently about the mod $\ell$ Hecke rings $R_k$, and prove as an immediate corollary that any prime $\ell \geq 5$ must eventually’’ divide the index of $\mathbb{T}_k$ in the ring of integers of $\mathbb{T}_k \otimes \mathbb{Q}$.

This result was later extended by Jochnowitz to show that the analogous statement is also true for any powers of any prime $\ell \geq 5$.

Event contact: jonathan dot pakianathan at rochester dot edu