IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Let K/Q be a real quadratic field. Given an automorphic representation π for GL2/K, let As±(π) denote the plus/minus Asai transfer of π to an automorphic representation for GL4/Q. In this paper, we construct a rigid analytic map from a subvariety of the universal eigenvariety of GL2/K to an eigenvariety of GL4/Q, which at nice classical points interpolate this Asai transfer. © 2019 Ramanujan Mathematical Society. All rights reserved.

Department of Mathematics

Journal of the Ramanujan Mathematical Society

P-adic Asai transfer

We establish a duality result proving the 'functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic Zp extension of a totally real number field. Further, we use this result to establish an 'algebraic functional equation' for the 'big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as a modest evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the 'big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the ℤ p 2 extension of an imaginary quadratic field. © 2017 International Press.

Fulltext

Asian Journal of Mathematics

Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

In this paper, we extend the endoscopic transfer of definite unitary group U(n), which sends a pair of automorphic forms of U(n1), U(n2) to an automorphic form of U(n1 + n2), to finite slope p-adic automorphic forms for definite unitary groups by constructing a rigid analytic map between eigenvarieties n1 × n2 → (n1+n2), which at classical points interpolates endoscopic transfer map. © 2016 Ramanujan Mathematical Society.

Endoscopic transfer between eigenvarieties for definite unitary groups

The aim of this book is to give a systematic exposition of results in some important cases where p-adic families and p-adic L-functions are studied. We first look at p-adic families in the following cases: general linear groups, symplectic groups and definite unitary groups. We also look at applications of this theory to modularity lifting problems. We finally consider p-adic L-functions for GL(2), the p-adic adjoint L-functions and some cases of higher GL(n).

World University root node

p-Adic Aspects of Modular Forms

Let K/F be a cyclic extension of odd prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow subgroup of the class group of K. In particular, when F contains the l-th roots of unity, we obtain bounds for the Fl -rank of the l-Sylow subgroup of K using genus theory. We obtain some results valid for general l. Following that, we obtain more complete, explicit results for l = 5 and F = ℚ(e2iπ/5). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (the latter is under GRH). We obtain explicit results in several cases. These results have a number of potential applications. For instance, some of them like Theorem 5.16 could be useful in the arithmetic of elliptic curves over towers of the form ℚ(e2iπ/5n, x1/5). Using the results on the class groups of the fields of the form ℚ(e2iπ/5, x1/5), and using Kummer duality theory, we deduce results on the 5-class numbers of fields of the form ℚ(x1/5). © 2015 Ramanujan Mathematical Society.

L-class groups of cyclic extensions of prime degree l

In this paper we show that the critical Eisenstein series of weight 2, E2critp, defines a smooth point in the eigen-curve C(l), where l is a prime different from p. We also show that E2critp,ordl defines a smooth point in the full eigencurve Cfull(l) and E2critp,ordl1,ordl2 defines a non-smooth point in the full eigencurve Cfull(l1l2). Further, we show that C(l) is étale over the weight space at the point defined by E2critp. As a consequence, we show that level lowering conjecture of Paulin fails to hold at E2critp,ordl. © Société Arithmétique de Bordeaux, 2015.