Seminar on number theory

Date and time: Wednesday, 18 January 2023 at 11:30 AM

Venue: Ramanujan Hall

Host: Dipendra Prasad

Speaker:  Rahul Dalal

Affiliation: Johns Hopkins University

Title: Explicit Trace Formulas and Statistics of Families of "Nice" Automorphic Representations

Abstract: The lecture will discuss a bit of the famous Arthur-Selberg trace formula with a view to applying it to families of automorphic representations. 

Mathematics Colloquium
Date and Time: Wednesday, 18 January, 2023, 4 pm
Host: Sudhir Ghorpade

Speaker: Sudesh Kaur Khanduja
Affiliation: Panjab University, Chandigarh

Title: When is Z[θ] the ring of integers?
Abstract: Let K = Q(θ) be an algebraic number field with θ an algebraic integer having minimal polynomial f(x) over Q. Let AK denote the ring of algebraic integers of K. In this talk, we shall discuss some necessary and sufficient conditions to be satisfied by f(x) so that AK = Z[θ]. In particular when f(x) is an irreducible trinomial x^n+ax^m +b ∈ Z[x], then we shall describe a set of necessary and sufficient conditions in terms of prime powers dividing a, b, m and n, for any prime p to divide the group index [A_K : Z[θ]]. Using the well known Dedekind Criterion, we shall also discuss a generalisation of this result for a simple ring extension R[η] of a valuation ring R to be integrally closed when η is a root of an irreducible trinomial x^n+ax^m +b belonging to R[x]. The latter result yields interesting number theoretic applications. This is partly based on joint works with A. Jakhar, B. Jhorar, Sumandeep Kaur, M. Kumar, and N. Sangwan.