Dipramit Majumdar

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.

The study of n-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves E1 and E2 over a number field K, Mazur–Rubin have defined them to be n-Selmer companion if for every quadratic character χ of K, the n-Selmer groups of E1χ and E2χ over K are isomorphic. Given a prime p, they have given sufficient conditions for two elliptic curves to be pr-Selmer companion in terms of mod-pr congruences between the curves. We discuss an analogue of this for Bloch–Kato pr-Selmer groups of modular forms. We compare the Bloch–Kato Selmer group of a modular form respectively with the Greenberg Selmer group when the modular form is p-ordinary and with the signed Selmer groups of Lei–Loeffler–Zerbes when the modular form is non-ordinary at p. We also indicate the relation between our results and the well-known congruence results for the special values of the corresponding L-functions due to Vatsal.

We determine the irreducible trinomials X3-aX + b for integers a,b which generate precisely all possible Galois extensions of degree 3 over ℚ. The proof, although involved, is elementary and one can parametrize all these polynomials explicitly. As an accidental by-product of the results, we prove that infinitely many primes congruent to 1 or-1 mod 9 are sums of two rational cubes-thereby, giving the first unconditional result on a classical open problem.

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 p2 extension of an imaginary quadratic field.