# Number Theory Abstracts Term 1, 2016-17

## A

- An arithmetic Chern-Simons invariant, by Minhyong Kim
Utilizing ideas of Dijkgraaf and Witten on 3D topological field theories, I will discuss a new invariant of Galois representations and present some calculations.

- Around Goldbach type problems, by Fernando Xuancheng Shao
This talk will be concerned about the ternary Goldbach problem of representing integers as sums of three primes, with these three primes restricted to a short interval. I will describe why it is difficult to apply the classical circle method in this sparse setting, and how to win by viewing the problem from a combinatorial perspective. One ingredient in the proof, rather surprisingly, involves understanding sets of doubling less than 4. This is joint work with Kaisa Matomaki and James Maynard.

## C

- Correlations of multiplicative functions, by Oleksiy Klurman
- We develop the asymptotic formulas for correlations \[\sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded ``pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences: first, we characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with bounded partial sums. This answers a question of Erd\H{o}s from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. Third, we discuss applications to the study of sign patterns of $(f(n),f(n+1),f(n+2))$ and $(f(n),f(n+1),f(n+2),f(n+3))$ where $f:\mathbb{N}\to \{-1,1\}$ is a given multiplicative function. If time permits, we discuss multidimensional version of some of the results mentioned above.

## D

- de Rham (\varphi, \Gamma)-modules and $p$-adic $L$-functions, by Joaquin Rodrigues Jacinto
Let p be a prime number. We will discuss how to associate, to a modular form of level N (possibly divisible by p), a $p$-adic $L$-function interpolating special values of the $L$ function of $f$. This construction is based on Kato's Euler system and on the theory of (\varphi, \Gamma)-modules. This comprises in particular the case of an elliptic curve with bad additive reduction at p. We will also discuss, using the $p$-adic Langlands correspondence and ideas of K. Nakamura, a functional equation on the Iwasawa theory for (\varphi, \Gamma)-modules of rank 2 and how this gives, on the one hand a functional equation for our $p$-adic $L$-function, and on the other hand results on Kato's local epsilon conjecture.

- Diagonal cycles and derivatives of Garrett $p$-adic $L$-functions, by R. Venerucci
Let $(\mathbf{f}, \mathbf{g},\mathbf{h})$ be a triple of Coleman families of cuspidal modular forms. Recently M. Greenberg and M. A. Seveso constructed triple product $p$-adic $L$-functions, interpolating the central values of the complex Garrett $L$-functions of the (balanced) classical specialisations of $(\mathbf{f}, \mathbf{g},\mathbf{h})$. After some motivation, I will discuss $p$-adic Gross--Zagier formulae for the Greenberg--Seveso $p$-adic $L$-functions, arising in the presence of an exceptional zero in the sense of Mazur--Tate--Teitelbaum. These formulae relate the derivatives of the $p$-adic $L$-functions to the $p$-adic Abel--Jacobi image of Gross--Kudla--Schoen diagonal cycles on triple products of Kuga--Sato varieties. This is joint work with M. Bertolini and M. A. Seveso.

## E

- Euler Systems from Special Cycles on Unitary Shimura Varieties and Arithmetic Applications, by Dimitar Jetchev
We construct a new Euler system from a collection of special 1-cycles on certain Shimura 3-folds associated to U(2,1) x U(1,1) and appearing in the context of the Gan--Gross--Prasad conjectures. We study and compare the action of the Hecke algebra and the Galois group on these cycles via distribution relations and congruence relations obtain adelically using Bruhat--Tits theory for the corresponding buildings. If time permits, we will explain some potential arithmetic applications in the context of Selmer groups and the Bloch--Kato conjectures for Galois representations associated to automorphic forms on unitary groups.

## O

- On L-functions attached to Jacobi forms of higher index, by Jolanta Marzec
Jacobi forms have been studied by several people and it has been known that they enjoy many similar properties to those possessed by Siegel modular forms. Therefore it is natural to ask whether the same holds for the associated L-functions (even though it is not known whether they may be identified with L-functions obtained from Galois representations). First of all: do they have meromorphic continuation and satisfy functional equation? can we say anything about their poles?

During the talk we will briefly introduce Jacobi forms and explain how one can use a doubling method to associate to them a (standard) L-function. We will present current knowledge on their properties and discuss challenges one have to face if (s)he works with Jacobi forms of higher index and non-trivial level. This is joint work with Thanasis Bouganis.

## S

- Stark points on elliptic curves and modular forms of weight one, by Alan Lauder
I shall discuss some work with Henri Darmon and Victor Rotger on the explicit construction of points on elliptic curves. The elliptic curves are defined over Q , and the points over fields cut out by Artin representations attached to modular forms of weight one.

- Subconvexity in certain Diophantine problems via the circle method, by Trevor Wooley
The subconvexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this subconvexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.

## T

- The Tate-Oort group scheme of order p, by Miles Reid
Three different group schemes of order

*p*defined over the prime field*FFp*play the role of the cyclic group*ZZ/p*in characteristic zero. The Tate-Oort group puts all of these together into a single family over a base of mixed characteristic. There are several forms of the construction, one quite elementary and one quite tricky, involving subtle and beautiful cyclotomic calculations.