Past Junior Number Theory Seminar

The notion of a pseudocharacter was introduced by A.Wiles for GL_2 and generalized by R.Taylor to GL_n. It is a tool that allows us to deal with the
deformation theory of a residually reducible Galois representation when the usual techniques fail. G.Chenevier gave an alternative theory of "determinants" extending that of pseudocharacters to arbitrary rings. In this talk we will discuss some aspects of this theory and introduce a similar definition in the case of the symplectic group, which is the subject of a forthcoming work joint with J.Quast. 

In this talk we will study the graph structure of supersingular isogeny graphs. These graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for their number of loops and multi-edges. We also find conditions under which these graphs are simple. To do so, we introduce a method of counting the total number of collisions (which are special endomorphisms) based on a trace formula of Gross and a known formula of Kronecker, Gierster and Hurwitz. 

The method presented in this talk can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we will see how this method was used to estimate a certain number of collisions and then show that isogeny graphs do not satisfy a certain cryptographic property that was falsely believed (and proven!) to hold.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor
function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is
to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their
conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this
talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on
the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

The Dedekind zeta function generalises the Riemann zeta
function to other number fields than the rationals. The analytic class number
formula says that the leading term of the Dedekind zeta function is a
product of invariants of the number field. I will say some things
about the class number formula, about L-functions, and about Stark's
conjecture which generalises the class number formula.

A pair of elliptic curves is said to be $N$-congruent if their mod $N$ Galois representations are isomorphic. We will discuss a construction of the moduli spaces of $N$-congruent elliptic curves, due to Kani--Schanz, and describe how this can be exploited to compute explicit equations. Finally we will outline a proof that there exist infinitely many pairs of elliptic curves with isomorphic mod $12$ Galois representations, building on previous work of Chen and Fisher (in the case where the underlying isomorphism of torsion subgroups respects the Weil pairing).

Isogeny-based cryptography is a candidate for post-quantum cryptography. The underlying hardness of isogeny-based protocols is the problem of computing endomorphism rings of supersingular elliptic curves, which is equivalent to the path-finding problem on the supersingular isogeny graph. Can path-finding be reduced to knowing just one endomorphism? An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this talk, we use the volcano structure of the oriented supersingular isogeny graph to take ascending/descending/horizontal steps on the graph and deduce path-finding algorithms to an initial curve. This is joint work with Sarah Arpin, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange and Ha T. N. Tran.

The Riemann hypothesis (RH) is one of the great open problems in
mathematics. It arose from the study of prime numbers in an analytic
context, and—as often occurs in mathematics—developed analogies in an
algebraic setting, leading to the influential Weil conjectures. RH for
curves over finite fields was proven in the 1940’s by Weil using
algebraic-geometric methods, and later reproven by Stepanov and
Bombieri by elementary means. In this talk, we use RH for curves to
prove the Weil bound for certain (Kloosterman) exponential sums, which
in turn is a fundamental tool in the study of prime numbers.

The Chabauty-Kim method is an effective algorithm for finding the $S$-integral points of hyperbolic curves by directly using the hyperbolicity in group-cohomological arguments. Central objects in the theory are affine spaces known as a Selmer schemes. We'll introduce the CK method and Selmer schemes, and demonstrate some additional structures possessed by Selmer schemes which can aid in implementing the CK method.
 

The Balog-Szemerédi-Gowers theorem is a powerful tool in additive combinatorics, that allows one to roughly convert any “large energy” estimate into a “small sumset” estimate. This has found applications in a lot of results in additive combinatorics and other areas. In this talk, we will provide a friendly introduction and overview of this result, and then discuss some proof ideas. No hardcore additive combinatorics pre-requisites will be assumed.

Modular forms are holomorphic functions on the upper half plane satisfying a transformation property under the action of Mobius transformations. While they are a priori complex-analytic objects, they have applications to number theory thanks to their connection with Galois representations. Weight one modular forms are special because their Galois representations factor through a finite quotient. In this talk, we will explain a different degeneracy: they contribute to the cohomology of a line bundle over the modular curve in degrees 0 and 1. We propose an arithmetic explanation for this: an action of a unit group associated to the Galois representation of the modular form. This extends the conjectures of Venkatesh, Prasanna, and Harris. Time permitting, we will discuss a generalization to Hilbert modular forms.

I'll introduce the classical arithmetic topology dictionary of Mumford-Manin-Mazur-Morishita-etc. I'll then present an interesting instance of parallel phenomena related to symplectic structures on moduli spaces of certain bundles. The arithmetic side turns out to be an application of Poitou-Tate duality. Depending on time, I'll delve into the delicate details which make the analogy useful for Diophantine geometers.

In this talk, I would like to discuss Deligne’s version of Geometric Class Field theory, with special emphasis on the correspondence between rigidified 1-dimensional l-adic local systems on a curve and 1-dimensional l-adic local systems on Pic with certain compatibilities. We should like to give a sense of how this relates to the OG class field theory, and how Deligne demonstrates this correspondence via the geometry of the Abel-Jacobi Map. If time permits, we would also like to discuss the correspondence between continuous 1-dimensional l-adic representations of the etale fundamental group of a curve and local systems.

Many classical arithmetic problems ranging from the elementary ones such as the density of square-free numbers to more difficult such as the density of primes, can be extended to integer matrices. Arithmetic problems over higher dimensions are typically much more difficult. Indeed, the Bateman-Horn conjecture predicting the density of numbers giving prime values of multivariate polynomials is very much open. In this talk I give an overview of the unfortunately brief history of integer random matrices.

The distribution of primes in arithmetic progressions (AP) s a central question of analytic number theory. It is closely connected to the additive behaviour of primes (for example in the Goldbach problem) and application of sieves (for example in the Twin Prime problem). In this talk I will outline the basic results without going into technical details. The central questions I will consider are: What are the different tools used to study primes in AP? In what ranges of moduli are they useful? What error terms can be achieved? How do recent developments fit into the bigger picture?

The moments of Hardy’s function have been of interest to number theorists since the early 20th century, and to random matrix theorists especially since the seminal work of Keating and Snaith, who were able to conjecture the leading order behaviour of all moments. Studying joint moments offers a unified approach to both moments and derivative moments. In his 2006 thesis, Hughes made a version of the Keating-Snaith conjecture for joint moments of Hardy’s function. Since then, people have been calculating the joint moments on the random matrix side. I will outline some recent progress in these calculations. This is joint work with Theo Assiotis, Benjamin Bedert, and Mustafa Alper Gunes.