• Speaker: Ashay Burungale (UT, Autstin)

• Title: On the arithmetic of elliptic curves

• Abstract: The talk will give an introduction to the arithmetic of elliptic curves defined over rational numbers, especially the conjecture of Birch and Swinnerton-Dyer, and some results.

• Speaker: Nikolaos Diamantis (Univ. of Nottingham)

• Title: Introduction to some aspects of the programme of Mazur and Rubin

• Abstract: We discuss a programme, due to Mazur and Rubin, which is motivated by the question of the behaviour of the Modell-Weil rank in abelian extensions of $\mathbb Q.$ This programme focuses on the statistical study of modular symbols and is structured around several conjectures about their mean and variance, the $\theta$-coefficients and vanishing questions for twisted L-functions of elliptic curves. We will describe these conjectures and outline progress towards their resolution.

• Speaker: Emmanuel Kowalski (ETH Zürich)

• Title: Equidistribution and L-functions

• Abstract: Some of the most interesting and important theorems in number theory can be phrased as equidistribution statements for arithmetic objects. The talk will survey some of the most classical equidistribution results, starting from Weyl's results on distribution of sequences, with a special focus the results which have a deep link with L-functions, from Dirichlet's Theorem on primes in arithmetic progressions to the Chebotarev Density Theorem and Deligne's equidistribution theorem.

• Speaker: Kohji Matsumoto (Nagoya Univ.)

• Title: The theory of multiple zeta-functions: the history and recent developments

• Abstract: The theory of multiple zeta-functions has a long history, from the days of Euler. However the big developments have begun in these several decades, mainly with the work of Hoffman, Zagier and Witten around 1990. The aim of this talk is to give a survey on this theory, starting with Euler’s classical computations and then mentioning various modern results, with focusing on the speaker’s recent theory on zeta-functions of root systems.

• Speaker: Frank Thorne (Univ. of South Carolina)

• Title: An Overview of Number Field Counting

• Abstract: How many number fields are there, of fixed degree and Galois group, and bounded discriminant?

I will give an overview of what is believed and what is known about this question, with an emphasis on recent and ongoing work. The question is notable for the wide range of tools that have proved useful, and I will attempt to give an audience a sense for the diversity and breadth of contemporary work.

• Speaker: Ashay Burungale (UT, Autstin)

• Title: Quantitative non-vanishing of Dirichlet L-values modulo p

• Abstract: Let p be an odd prime and N a positive integer prime to p. We discuss quantitative results on the number of Dirichlet characters modulo N with corresponding critical L-values being indivisible by p, as N varies (joint with Hae-Sang Sun).

• Speaker: Dohoon Choi (Korea Univ.)

• Title: The algebraic parts of the central values of quadratic twists of modular $L$-functions modulo l

• Abstract: In this talk, I will talk about non-vanishing of central values of modular L-functions twisted by quadratic characters.

• Speaker: Nikolaos Diamantis (Univ. of Nottingham)

• Title: L-series of weakly holomorphic forms and applications

• Abstract: A new L-series recently attached to weakly holomorphic forms (jointly with Lee, Raji and Rolen) is presented and some of its basic properties outlined. Applications to cycle integrals and computational questions are discussed.

• Speaker: Chan-Ho Kim (KIAS)

• Title: Understanding Selmer groups of modular forms via modular symbols

• Abstract: This is work in progress. We discuss how Bloch-Kato Selmer groups of modular forms can be understood in terms of modular symbols. We do not impose any ordinary or Fontaine-Laffaille type assumption.

• Speaker: Shinichi Kobayashi (Kyushu Univ.)

• Title: The p-adic valuation of local resolvents and anticyclotomic Hecke L-values of imaginary quadratic fields at inert primes

• Abstract: We explain an asymptotic formula for the p-adic valuation of Hecke L-values of an imaginary quadratic field at an inert prime p along the anticyclotomic Zp-tower. The key is determination of the p-adic valuation of local resolvents in Zp-extensions.

• Speaker: Emmanuel Kowalski (ETH Zürich)

• Title: New L-functions for general Fourier transforms

• Abstract: (Joint work with A. Forey and J. Fresán) The exponential sums over finite fields that are parameterized by characters of a commutative algebraic group can be combined into a kind of Euler product, similarly to L-functions. When considering additive characters of the finite field, this function is related to Deligne's Fourier transform. In other cases, such as multiplicative characters, it is however not an L-function in the usual sense. The talk with explain the definition of these objects, and present examples as well as some first applications and open questions.

• Speaker: Jungin Lee (KIAS)

• Title: Mixed moments and the joint distribution of random groups

• Abstract: The moment problem is to determine whether a probability distribution is uniquely determined by its moments. Recently, the moment problem for random groups has been applied to the distribution of random groups, in particular the cokernels of random p-adic matrices. In this talk, we introduce the mixed moments of random groups and apply this to the joint distribution of random abelian and non-abelian groups. In the abelian case, we provide three universality results for the joint distribution of the multiple cokernels for random p-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations.

• Speaker: Min Lee (Univ. of Bristol)

• Title: An extension of converse theorems to the Selberg class

• Abstract: : The converse theorem for automorphic forms has a long history beginning with the work of Hecke (1936) and a work of Weil (1967): relating the automorphy relations satisfied by classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. The classical converse theorems were reformulated and generalized in the setting of automorphic representations for GL(2) by Jacquet and Langlands (1970). Since then, the converse theorem has been a cornerstone of the theory of automorphic representations. Venkatesh (2002), in his thesis, gave new proof of the classical converse theorem for modular forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this talk, we extend Venkatesh’s proof of the converse theorem to forms of arbitrary levels and characters with the gamma factors of the Selberg class type. This is joint work with Andrew R. Booker and Michael Farmer.

• Speaker: Yoonbok Lee (INU)

• Title: Value distribution of L-functions

• Abstract: We introduce recent progress on the joint value distribution of L-functions.

• Speaker: Kohji Matsumoto (Nagoya Univ.)

• Title: Dirichlet series related with Goldbach counting functions —— The classical case and the function-field analogue

• Abstract: We consider the Dirichlet series whose numerator is given by the Goldbach counting function, and discuss its analytic properties. First, we report our former result in the classical case, which says that such series seems to have the natural boundary, and it is possible to prove the existence of such boundary under some plausible conjectures. Then we report our new result in the function-field case. In this case we can show the condition, without any unproved hypothesis, when the series has the natural boundary. This is a joint work with Shigeki Egami.

• Speaker: Kentaro Nakamura (Saga Univ.)

• Title: Construction of a p-adic L-function over the Coleman-Mazur eigencurve

• Abstract: The Coleman-Mazur eigencurve is a rigid analytic curve which parametrizes (overconvergent) modular forms which are non-supercuspidal at p. In this talk, we first construct a zeta element for the Galois representation over the eigencurve using the zeta element for the corresponding universal deformation which was constructed by the speaker. Applying a generalization of Perrin-Riou theory to this zeta element, we construct a p-adic L-function over the eigencurve. This is joint work with Chan-Ho Kim (KIAS).

• Speaker: Frank Thorne (Univ. of South Carolina)

• Title: Error Terms in Arithmetic Statistics

• Abstract: I will give an overview of an ongoing program (with many collaborators) to obtain error terms in a variety of arithmetic statistics results which are quantitatively as strong as possible. I will describe some of the techniques that go in the proofs, as well as some of the applications that one can obtain.