SCIENTIFIC PROGRAMS AND ACTIVITIES
|February 21, 2024
R. Balasubramanian (IMSc)
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
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
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.
We discuss some recent studies into logarithmic derivatives of
Artin L-functions at $s=0$ and the conjectural relationship to periods
of Abelian varieties.
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.
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.
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
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.
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.
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.
We will survey some techniques involved in the evaluation of infinite sums of the form
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.