ICTS

The link between automorphic forms and isogeny height estimates lies in their connection to elliptic curves, and to abelian varieties more generally. Automorphic forms are mathematical functions that have symmetry properties under certain groups, such as the modular group. They are closely related to elliptic curves and to the arithmetic of modular curves, over global fields, local fields, finite fields.

Isogenies are morphisms between elliptic curves that preserve certain algebraic properties. They are a key feature in the definition of modular curves.

The height of an elliptic curve (its Faltings height, or the height of its modular j-invariant for instance) measures its complexity and is related to some of its arithmetic properties. Isogeny height estimates are of several types: they give a way to control the degree of a minimal isogeny existing between isogenous elliptic curves, or quantify the difference in heights between isogenous elliptic curves. Isogeny height estimates have applications in various areas, including results in Diophantine geometry. They help in understanding the complexity of modular curves, via the study of their Faltings height, of the auto-intersection of their dualizing sheaves, and of modular polynomials.

The course will consist of two talks and will be focusing on two main themes:

1. Height theory: how to measure the size of objects in algebraic geometry?
2. Measuring the size of modular curves.

The link between automorphic forms and isogeny height estimates lies in their connection to elliptic curves, and to abelian varieties more generally. Automorphic forms are mathematical functions that have symmetry properties under certain groups, such as the modular group. They are closely related to elliptic curves and to the arithmetic of modular curves, over global fields, local fields, finite fields.

Isogenies are morphisms between elliptic curves that preserve certain algebraic properties. They are a key feature in the definition of modular curves.

The height of an elliptic curve (its Faltings height, or the height of its modular j-invariant for instance) measures its complexity and is related to some of its arithmetic properties. Isogeny height estimates are of several types: they give a way to control the degree of a minimal isogeny existing between isogenous elliptic curves, or quantify the difference in heights between isogenous elliptic curves. Isogeny height estimates have applications in various areas, including results in Diophantine geometry. They help in understanding the complexity of modular curves, via the study of their Faltings height, of the auto-intersection of their dualizing sheaves, and of modular polynomials.

The course will consist of two talks and will be focusing on two main themes:

1. Height theory: how to measure the size of objects in algebraic geometry?
2. Measuring the size of modular curves.

I will discuss some problems in Diophantine approximation which can be studied via the ergodic theory of group actions on homogeneous spaces of Lie groups. No prior knowledge of homogeneous dynamics will be assumed.

For a self-dual cuspidal automorphic representation of GL(2) over a number field, not of solvable polyhedral type, we obtain lower bounds on the upper Dirichlet density of the set of primes at which the Hecke eigenvalues exceed a given parameter t. We will also discuss the joint behaviour of two such non-twist-equivalent representations, bounding the occurrence of primes at which the Hecke eigenvalues of the first representation are greater than those of the second.

This mini course is an introduction to Eisenstein Cohomology and applications to the special values of automorphic L-functions. We will discuss two applications mostly via examples: (1) denominators of Eisenstein classes are related to special values of L-functions. (2) congruences for modular forms leading to congruences between ratios of special values of Rankin-Selberg L-functions.

The classical Selberg trace formula and its automorphic versions like Arthur-Selberg trace formula have played a big role in Number theory over last many decades. In this talk, we give an overview of motivations and examples of trace formula in mathematics including the Selberg trace formula for compact quotients. We also discuss about various results on different types of spectra of locally symmetric space that use Selberg trace formula. This is part of my joint work(s) with C.S. Rajan, Supriya Pisolkar, Gunja Sachdeva, Kaustabh Mondal over last many years.

This mini course is an introduction to Eisenstein Cohomology and applications to the special values of automorphic L-functions. We will discuss two applications mostly via examples: (1) denominators of Eisenstein classes are related to special values of L-functions. (2) congruences for modular forms leading to congruences between ratios of special values of Rankin-Selberg L-functions.

In this talk, we wish to outline a method to compute a lower bound for the number of non-CM Galois orbits of newforms in $\mathcal{S}_k(N,\Psi)$ with non-trivial quadratic nebentypus $\Psi$ for sufficiently large weights. This is a joint work with S. Das, D. Das, T. Mandal and S. Mondal.