I am co-organizing the Junior Number Theory Seminar with Nischay Reddy. We will be meeting on Fridays 2-3pm in BAB024. If you are interested in giving a talk, please reach out to Nischay or myself!

This is the continuation of a past seminar organized by University of Toronto number theory students.

• September 16, 2022 - Malors Espinosa Lara

The Langlands Program and Orbital Integrals

Abstract: In this talk I will talk about an object that appears in the Langlands program and that is elusive in its manifestation and understanding. I will introduce the Orbital Integral, which are main players in the Langlands Program and yet do not yield to easy computation nor understanding and much depends on it.

• September 23, 2022 - Ila Varma

Ray Class Group Asymptotics of Quadratic Fields

Abstract: In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn's theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over Q. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn's asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields with fixed conductor.

• September 30, 2022 - Enrique Nuñez Lon-Wo

Bounding the Upper Density of the Eisenstein Set

Abstract: We find an upper bound on the upper density of the Eisenstein set of d such that d is congruent to 5 modulo 8 and x^2 - dy^2 = 4 has no odd integer solutions in set of d congruent to 5 modulo 8 using theorems discussed in Ila's talk and the related paper. In particular, we find a lower bound on the lower density of a positive square-free integer congruent to five modulo eight not being in the Eisenstein set. In the process we prove a general statement which allows us to bound lower and upper densities of a set X over Y given that the average value of X exists.

• October 7, 2022 - Alexander Slamen

Bird's Eye View on Arithmetic Statistics

Abstract: After hearing two talks on specific work in arithmetic statistics, I will take us a step backwards and take a bird's eye view of the subject. We'll discuss landmark results, questions of interest, and important techniques.

• October 14, 2022 - Sasha Shmakov

Cohomology of Shimura Varieties (and more Arithmetic Statistics)

Abstract: A question of fundamental importance in arithmetic geometry is how to compute the cohomology of local systems on Shimura varieties, and closely related locally symmetric spaces, and to extract Galois representations from this procedure. I will try to summarize how this procedure goes with focus on two examples: modular curves, which parameterize elliptic curves with extra structure, and Siegel modular 3-folds, which parameterize Abelian surfaces with extra structure. If time permits I will try to explain how this can be applied to questions about arithmetic statistics for varieties over finite fields.

• October 21, 2022 - Cancelled

• October 28, 2022 - Peter Angelinos

What is p-adic integration anyway?

Abstract: People often ask me what p-adic integration is, perhaps because it sounds impressive and mysterious. When I say it is a fancy version of counting, no one believes me. But, thanks to a theorem of Weil, it is indeed simply a point-count. Weil proved that the volume of a smooth variety over a p-adic field is related to the number of solutions of its defining polynomials. I will discuss this theorem and its application (via the Weil Conjectures) to mirror symmetry by Batyrev. Time permitting, I will discuss how this relates to my research.

• November 4, 2022 - Jiyuan (Maki) Lu

TBA

• November 18, 2022 - Hymn Chan

TBA

• November 25, 2022 - Matt Olechnowicz

TBA

• December 2, 2022 - Nischay Reddy

TBA