UA Grad ANT

26Fa UArizona Grad Algebra & Number Theory Seminar

In spring 2026, the UA Grad ANT Seminar is organized by Tyler Kline. This website is maintained by Napoleon Wang.

This is a weekly seminar for graduate students in algebra, related fields (algebraic geometry, arithmetic geometry, algebraic topology, algebraic combinatorics, etc.) and number theory. It is an ideal environment to practice public speaking skills, connect with peers, and explore new mathematical ideas. Presentations do not need to be novel research. Participants are welcome to share any algebra or number theory topic that currently interests them.

We meet Fridays, 11 AM-12 PM Arizona Time, in ENR2 S395, unless otherwise noted.

If you want to be added to the mailing list, feel free to contact Tyler at tylerkline@arizona.edu.

Participants can sign up for talks directly in the sheet below.

Grad Student Algebra and Number Theory Seminar - Fall 2026

Upcoming Talk(s)

Oct 16 Noah Smith
Title: Modular Forms and their Hecke Operators Mod 2
Abstract: Modular forms are a central object in modern number theory. The coefficients of their Fourier expansions and the algebra of Hecke operators acting on them are closely related to various problems in number theory, for example in the study of Galois representations, elliptic curves, and quadratic forms. This talk will explore what happens when we reduce these Fourier expansions mod 2. We will primarily follow two papers of Nicolas and Serre which describe the Hecke algebra acting on these mod-2 modular forms, in particular showing that the Hecke operators mod 2 are generated as a power series ring by $T_3$ and $T_5$ and act nilpotently with predictable nilpotency orders. If time permits, we will discuss some surprising connections with the parity of the partition function.

Previous Talks (with notes if applicable)

Oct 9 Sean Zhu
Title: Toric Exponential Sums in Families
Abstract: Exponential sums connect point counting over finite fields with $L$-functions. I will introduce the cohomological approach of Adolphson and Sperber, then explore lower-order parameter deformations, where a fixed cohomological basis lets us study how Frobenius and the local $L$-functions vary.

Oct 2 Xinran Qian
Title: Hurwitz class number formula
Abstract: The Hurwitz class number, which measures the size of equivalence class of positive definite binary quadratic forms, is one of the central objects in algebraic number theory. The Hurwitz formula is a miraculous identity that gives an elementary expression for a certain sum of Hurwitz class numbers, and it was established by Hurwitz in 1885 using the modular j-function and the modular polynomial.
In 1993, Gross and Keating gave a geometric proof of the Hurwitz formula by finding geometric intersection number of modular correspondences, using the moduli description of modular curves and the theory of complex multiplication. They also found a remarkable generalization for arithmetic intersection numbers of modular correspondences, relating it to the central derivative of Siegel Eisenstein series on Sp(6).
On Friday, I will try to present what I understand from the paper by Gross-Keating and connect it to some recent developments in the Kudla program.

Sep 25 Ameya Borwankar
Title: Vector Bundles on Curves
Abstract: The twistor plays a fundamental role in both four-dimensional space-time geometry and Number Theory. In the framework of algebraic geometry, we can define it as an abstract curve. In this talk, we shall study the classification of vector bundles over abstract curves such as the projective line P^1 and the twistor using slope formalism and Harder-Narasimhan Theory.

Sep 11 Tyler Kline
Title: t-adic L-functions of Exponential Sums
Abstract: Dwork’s approach to proving the rationality of the zeta function for a non-singular projective variety was to produce a particular operator acting on a p-adic Banach space which gives a trace formula that determines the zeta function. For the last 60 years, mathematicians have been building off of Dwork’s methods to study the closely related L-functions of exponential sums. One family of objects are the L-functions associated to “twisted” exponential sums, where multiplicative characters are introduced into the exponential sums. In this talk, we will review some foundational results in the untwisted case and discuss how they are extended to the twisted case from a p-adic analytic perspective.