Number Theory Tea 2020

Currently, we are meeting on Mondays 4:30PM Princeton time at https://princeton.zoom.us/j/92542250798

Upcoming talks

12/14: No speaker

12/21: No speaker

12/28: No speaker

Past talks

7/7: Evan O'Dorney, *Old Nassau*

Karl Langlotz wrote the memorable music of our alma mater, "Old Nassau," in an unassuming residence on Mercer Street in 1859. I invite you to peruse the sheet music and explore its enduring excellence, much of which is owed to musical wisdom and techniques that had been honed for centuries before. There is a lot even in a short song!

7/14: Gyujin Oh, *Differential equations and transcendence*

I will sketch a very likable, fairly motivated proof of some transcendence results (including transcendence of π). It uses a systematic study of differential equations satisfied by Taylor series with coefficients in a number field.

7/21: Linus Hamann, *Deformation of local systems and geometric Eisenstein series*

We will discuss the theory of Eisenstein series in the context of classical Langlands over function fields as well as the geometric Langlands program, following ideas of Braverman, Gaitsgory, and Laumon. At the end, we will explore the relation between these two objects in the context of deforming a GL_{2}-local system induced from the maximal torus, vaguely hinting at an analogy between this situation and some ideas in the usual arithmetic Langlands program.

7/28: Evan O'Dorney, *MAZE: Solve the World's Most Challenging Puzzle (1986 book)*

After the past lectures on serious mathematics, here is some light relief. In 1985, children's book illustrator Christopher Manson published a book MAZE: Solve the World's Most Challenging Puzzle. The publisher (Holt, Rinehart & Winston) also organized a contest promising $10,000 for a winning solution to the Maze. It was not known until recently whether any money was awarded. I invite you to browse the book, which has been digitized on the site intotheabyss.net, and join me in solving at least one of the maze's three parts.

8/4: Eric Chen, *Path integrals*

The path integral approach to quantum mechanics is one of the most conceptually transparent ways to describe quantum particles; however, the physical quantities one wishes to compute are trapped in complicated integrals. In this talk I will describe how the problem of estimating these integrals links some interesting mathematical ideas. No physics background is required if you just trust me on some things.

8/11: Conling Qiu, *Diophantine approximation on abelian varieties and Shimura varieties*

This is a very large topic, I will choose some some results and conjectures which I think are interesting. It will be more on algebraic number theory, rather than analytic theory. It is good enough if you know elliptic curves and modular curves.

8/18: Leo Lai, *p-adic L-functions*

I will give an introductory talk on p-adic L-functions, their special values, and arithmetic significance. Linear forms in p-adic logarithms will make a guest appearance.

8/25: No speaker

8/31: Jack Sempliner, *Commutative Subrings of Noncommutative Rings*

In the late 1970s, Igor Krichever was working on the Zaharov-Shabat method of "inverse scattering," a technique for integrating certain nonlinear PDE. In his work he discovered a remarkable dictionary between two seemingly very disparate types of data: on the one hand one has (1) a commutative subring of an appropriate noncommutative ring of differential operators; on the other one has (2) a suitable algebraic curve, a vector bundle over it, and some points on the curve. The dictionary has profound applications, first to constructing solutions to the KdV equation, but later to number theory and the Langlands Correspondence in positive characteristic. We'll talk about this amazing dictionary and give some idea of how it works.

If people are curious about prerequisites all that will be required is some understanding of line bundles on projective curves.

9/7: Bat Dejean, *Model theory of local fields*

Model theory is a branch of logic studying the first-order properties of “sets with additional structure” in a fairly general sense. Classical work by Tarski gives a fairly complete account of the model theory of C and R. In a series of 1965-66 papers, Ax and Cohen gave analogous results for Qp. Their most remarkable theorem, however, is a principle for directly transferring results between Qp and Fp((t)). This gave a partial positive result on a conjecture by Artin in Diophantine analysis which later turned out to be false.

After giving some basic model-theoretic background, the majority of the talk will be dedicated to algebraically closed fields because they are easy to work out in detail, easy to relate to algebraic geometry, and analogous to the real and p-adic cases. I will then say some things about what work is necessary to do the real and p-adic cases, then state and sketch Ax and Kochen's cross-characteristic transfer theorem.

9/14: Sameera Vemulapalli, *Geometry of cubic rings*

To a cubic ring, one can associate a plane quartic and a plane cubic with specific singularities. These types of quartics were studied extensively by classical algebraic geometers, and are interesting in their own right. Additionally, the geometry of these curves (singularities, flexes, etc.) contains interesting arithmetic data about the cubic ring.

9/21: Leo Lai, *Cross-characteristic transfer for the fundamental lemma*

We will extend the transfer results from Bat's talk to a much broader class of theorems using Cluckers-Loeser's motivic integration. This in particular includes the fundamental lemma.

9/28: Marco Sangiovanni, *Decomposition groups are big*

Let $S$ be a finite set of finite places of $\Q$ and consider $\Q_S$ the maximal extension of $\Q$ that is unramified outside $S$ and $G_S$ its Galois group. A consequence of Minkowski’s discriminant bound is that, if $S$ is empty, $\Q_S = \Q$. However, If $S$ is non empty, the structure of $G_S$ is poorly understood, despite these groups arising very often in arithmetic geometry. Ralph Greenberg asked whether one could describe the decomposition groups of $G_S$ - the overlying conjecture being that these local groups are as « big » as the ramification allows. I will present some results of Chenevier-Clozel that confirm this conjecture.

10/5: No speaker

10/12: Trajan Hammonds, *Galois representations attached to modular forms*

There's a well known result of Deligne (and Serre) that associates modular forms to Galois representations. I will sketch this construction in the weight 2 case.

10/19: Tuan Do, *Classic arithmetic results about partition number p(n)*

This talk will be about the Ramanujan congruences for partition function, e.g. for every n, p(5n+4) is congruent to 0 mod 5. I will give different proofs (from standard, random to overkill) for this fact. Note: This year marks the 100th death anniversary of Ramanujan.

10/26: Katy Woo, *Twin primes and Chowla's conjecture for function fields*

I will talk about the analytic number theory aspects of the recent proof by Sawin and Shusterman of twin primes and Chowla's conjecture in the function field setting.

11/2: Ashvin Swaminathan, *Counting points in cuspidal regions*

In this talk, we present a new technique for counting the number of SL_2(Z)-classes of reducible binary cubic forms having bounded discriminant. Although this problem admits a solution using purely elementary methods, the benefit of the new technique is that it applies to counting reducible orbits in a whole host of other representations.

11/9: Gyujin Oh, *A new proof of the Neron-Ogg-Shafarevich criterion*

The Neron-Ogg-Shafarevich criterion says that good reduction of elliptic curve over a local field can be read off from its Tate module. We will discuss an analogous criterion for curves of higher genus, where one uses fundamental groups, as opposed to Tate modules (pi_1 vs. pi_1^ab=H_1). It's proved by reducing to the case of Riemann surfaces (!).

11/16: Gyujin Oh, *Nonabelian Neron-Ogg-Shafarevich criterion*

The Neron-Ogg-Shafarevich criterion says that good reduction of elliptic curve over a local field can be read off from its Tate module. We will discuss an analogous criterion for curves of higher genus, where one uses fundamental groups, as opposed to Tate modules (pi_1 vs. pi_1^ab=H_1). It's proved by reducing to the case of Riemann surfaces (!).

11/23: Boya Wen, *A Gross-Zagier Formula for CM cycles over Shimura Curves*

In this talk I will introduce my thesis work in progress to prove a Gross-Zagier formula for CM cycles over Shimura curves. The formula connects the global height pairing of special cycles in Kuga varieties over Shimura curves with the derivatives of the L-functions associated to weight-2k modular forms. As a key original ingredient of the proof, I will introduce some harmonic analysis on local systems over graphs, including an explicit construction of Green's function, which we apply to compute some local intersection numbers.

11/30: Raul Alonso Rodriguez, *Big Heegner points*

In his paper "Variation of Heegner points in Hida families", Howard constructed classes in the cohomology of the Galois representation attached to a Hida family interpolating the Kummer images of the Heegner points corresponding to each weight two specialisation. These classes, which he calls "big Heegner points", yield an anticyclotomic Euler system. In this talk we will present Howard's construction.

12/7: Jack Sempliner, *The orders of finite subgroups of rational points of reductive algebraic group*

We'll talk about optimal bounds on the size of finite subgroups of G(K), where G is a reductive algebraic group and K is a number field. Specifically we will reprove classical results due to Minkowski and Schur which give provably optimal bounds for Gl_n, and, time permitting, move on to talk about generalizations of these bounds to arbitrary G.