# SCIENTIFIC PROGRAMS AND ACTIVITIES

September 21, 2017

## July 6-9, 2011 Conference on Transcendence and L-functions Fields Institute to be held at the Fields Institute, Toronto (map)

Organizers: M. Ram Murty (Queen’s) and V. Kumar Murty (Toronto)

Abstracts

R. Balasubramanian (IMSc)
Catalan's conjecture in imaginary quadratic fiel

In 1844 Eugene Charles Catalan conjectured that equation $x^m-y^n=1$ has only two nontrivial solutions $(x,y,m,n)$ in integers, viz, $(\pm 3, 2, 2,3)$. An essential ingredient in this direction was developed by J. W. S. Cassels, who proved that when $m$ and $n$ are odd primes then any solution $(x,y,m,n)$ satisfies $m|y$ and $n|x$. Equipped with Cassels' criterion Preda Mihailescu used the theory of cyclotomic fields (annihilator of class groups) in ingenious way to settle the problem in 2002. We have obtained Cassels' criteria when $x$ and $y$ are allowed to take values in the ring of integers of an imaginary quadratic fields with class number $1$. Depending on time we may highlight on further strategy for solving Catalan's conjecture for these number fields.

___________________________
Sanoli Gun (IMSc)
Non-vanishing of L(k,f) for k>1, irrationality and multiple zeta values

In this talk, we discuss the non-vanishing of L(s,f) at integers greater than 1, where f is a rational valued periodic function. These questions are linked to irrationality of zeta values and multiple zeta values.

___________________________
Manfred Kolster (McMaster)
Special values of Dedekind zeta-functions and motivic cohomology

In this (mainly expository) talk we will concentrate on the algebraic part of the special values of Dedekind zeta-functions at negative integers. The conjectural interpretation of these non-zero rational numbers in terms of orders of algebraic K-theory groups goes back to Lichtenbaum in the 1970's. We will describe the known results and the basic techniques used in approaching
the conjectures. The conjectures were only formulated up to powers of 2, and we will discuss how recent work of Voevodsky allows to remove this "de ciency" by replacing K-theory by motivic cohomology.

___________________________
Growth of logarithmic derivatives of Dirichlet L-functions

This talk is based on a joint work with Prof. V.Kumar Murty. It consists mainly of two theorems. In Theorem 1 we wil prove that for infinitely many fundamental discriminants D we have :

|-L'/L(1,\chi_D)| >> loglog |D|,

which is an analogue of a classical result of Chowla for L(1,\chi_D. Here, \chi_D is the real character of discriminant D. On the other hand, we will show in Theorem 2 that despite the Omega Theorem itself, the average over fundamental discriminants of L'/L(1,\chi_D)$will be a constant. ___________________________ Kumar Murty (Toronto) Periods and Special values of Logarithmic derivatives of L-functions We discuss some recent studies into logarithmic derivatives of Artin L-functions at$s=0$and the conjectural relationship to periods of Abelian varieties. ___________________________ Ram Murty (Queen's) Transcendence of Special Values of L-series We will give a brief survey of the transcendence of zeta and L-series at integer arguments. We focus on special values of Artin L-series and discuss theorems and conjectures related to them. ___________________________ A. Raghuram (Oklahoma) Special values of automorphic L-functions I will give an introduction to a class of periods attached to automorphic forms which arise via a comparison of rational structures on various models (for example, Whittaker models, Shalika models, cohomological models, etc.) of the underlying automorphic representation. These periods capture the possibly trascendental parts of the special values of certain automorphic L-functions. For much of my talk, I will deal with these themes in the context of Manin's and Shimura's classical results on the special values of L-functions attached to (Hilbert) modular forms. Towards the end I will talk of several possible generalizations to the context of Rankin-Selberg L-functions for GL(a)xGL(b). This last part of my talk will touch upon the results of several projects of mine, some of which are joint projects with Harald Grobner, Guenter Harder, Freydoon Shahidi and Naomi Tanabe. ___________________________ Purusottam Rath (CMI) Non-vanishing of L(1,f), the state-of-the-art We give an overview of the question of non-vanishing of periodic L- functions L(s,f) at s = 1. ___________________________ Damien Roy (Ottawa) Algebraic independence and small value estimates An estimate for the transcendence degree of a family of numbers has three parts. The first is analytic and consists in the construction of a sequence of auxiliary functions taking small values in the field K generated by these numbers over Q. The second is arithmetic. It aims at showing that these family of numbers (one for each auxiliary function) are eventually all zero, assuming that the transcendence degree of K is small. Typically this uses Philippon's criterion for algebraic independence. The third part is geometric. It is a zero estimate. Based on the vanishing obtained in the second part, it leads to a final contradiction which proves the desired lower bound for the transcendence degree of K. The purpose of a small value estimate is to combine the last two steps in order to make a better use of the data coming out of the construction of the auxiliary function. We discuss the philosophy behind it with some examples, and present a recent result related to algebraic group Ga x Gm. The central ingredient needed for this is a multiplicity estimate for the resultant of polynomials in several variables. ___________________________ Chris Smyth (Edinburgh) Zeros of Ramunajan and related polynomials In joint work with Ram Murty and Rob Wang, we study the zeros of Ramanujan polynomials. This enables us to express$\zeta(n)$for odd$n\$ at least 9 as a difference of Eichler integrals evaluated at some of these zeros. We also consider the zeros of a closely related polynomial, studied by Lalin and Rogers. Our results have implications for the values of Eichler integrals at these zeros.

___________________________
Cameron Stewart (Waterloo)
Transcendence, linear forms in logarithms and applications

In this talk we shall discuss the train of ideas that led from Gelfond and Schneider's solution of Hilbert's 7th Problem to estimates for linear forms in the logarithms of algebraic numbers and to a multitude of applications. We shall also discuss p-adic aspects of the theory.

___________________________
Michel Waldschmidt (Paris VI)
Transcendence properties of Euler's Gamma and Beta functions

In the first part of the lecture, we introduce the history and the state of the art of the subject: we give a survey of known transcendence results related with Gamma and Beta values, from the early results due to C.L.~Siegel (1931) and Th.~Schneider (1934) to the more recent algebraic independence results by Yu.~V.~Nesterenko (1996). One of the important tools, beside transcendence methods, is the formula of Chowla-Selberg.

In the second part of the talk, we discuss open problems, from the conjectures of D.~Rohrlich and S.~Lang, to Y.~André's Galois theory for some transcendental numbers, including the periods of M.~Kontsevich and D.~Zagier.

___________________________
Chester Weatherby (Delaware)
Transcendence of sums over the integers

We will survey some techniques involved in the evaluation of infinite sums of the form

$$\sum_{n \in {\Bbb Z}} f(n){A(n) \over B(n)}$$

where f is periodic and A(x), B(x) are polynomials and summation is taken over Z. Once we discuss evaluation of these sums in a general setting, we are able to characterize the transcendental nature of these numbers in various somewhat natural settings.