I am a Hans Rademacher instructor in the Mathematics Department at the University of Pennsylvania. I completed my PhD at the University of Washington. My thesis advisor is Ralph Greenberg.

My research interests in algebraic number theory are motivated by questions in Iwasawa theory and Hida theory.

My CV

On Selmer groups and factoring p-adic L-functions,

accepted for publication in IMRN (Journal) (Arxiv)Height one specializations of Selmer groups,

submitted : (Arxiv) (Sage Worksheet)On free resolutions of Iwasawa modules (Preprint)

## Show/Hide Abstract

Let $\Lambda$ (isomorphic to $\mathbb{Z}_p[[T]]$) denote the usual Iwasawa algebra and $G$ be the Galois group of a finite Galois extension $L/K$ of totally real fields. The main theorems in this article describe the precise conditions under which non-primitive Iwasawa modules, over the cyclotomic $\mathbb{Z}_p$-extension, have a free resolution of length one over the group ring $\Lambda[G]$. As one application, under these conditions of the main theorems, the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive $p$-adic $L$-function (which is an element of a $K_1$-group) in a maximal $\Lambda$-order. As another application, we consider an elliptic curve over $\mathbb{Q}$ with a cyclic isogeny of degree $p^2$. We relate the characteristic ideal, in the ring $\Lambda$, of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over $\Lambda$, associated to two non-primitive classical Iwasawa modules.

On second Chern classes and tensor product of Hida families (in preparation)

Samit Dasgupta has proved a formula factoring a 3-variable Rankin-Selberg $p$-adic $L$- function as a product of a 2-variable $p$-adic $L$-function associated to the adjoint representation of a Hida family and a Kubota-Leopoldt $p$-adic $L$-function. We prove a result involving Selmer groups that along with Dasgupta's result is consistent with the main conjectures associated to the 4-dimensional representation (to which the 3-variable Rankin-Selberg $p$-adic $L$-function is associated to), the 3-dimensional adjoint representation and the 1-dimensional representation (to which the Kubota-Leopoldt $p$-adic $L$-function is associated to). Under certain additional hypotheses, we indicate how one can use work of Urban involving one-way divisibility for the adjoint r epresentation to deduce main conjectures for the adjoint representation and the 4- dimensional representation.

Dasgupta's method of proof is based on an earlier work of Gross in 1980. Gross's work involved factoring a 2-variable $p$-adic $L$-function associated to an imaginary quadratic field (constructed by Katz) into a product of two Kubota-Leopoldt $p$-adic $L $-functions. In 1982, Greenberg proved the corresponding result on the algebraic side involving classical Iwasawa modules, as predicted by the main conjectures for imaginary quadratic fields and $\mathbb{Q}$. Our methods are inspired by this work of Greenberg. One key technical input to our methods is studying the behavior of Selmer groups under specialization.

We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to 4-dimensional Galois representations coming from (i) the tensor product of two cuspidal Hida families $F$ and $G$ , (ii) its cyclotomic deformation (iii) the tensor product of a cusp form $f$ and $G$, where $f$ is a classical specialization of $F$ of weight $k \geq 2$. We first develop control theorems to relate (a) the Selmer group associated to the tensor product of Hida families $F$ and $G$ to the Selmer group associated to their cyclotomic deformation and (b) the Selmer group associated to the tensor product of $f$ and $G$ to the Selmer group associated to the tensor product of $F$ and $G$. On the analytic side of the main conjectures for these 4-dimensional representations, Hida has constructed 1-variable, 2-variable and 3-variable Rankin-Selberg $p$-adic $L$-functions. Granting the validity of main conjectures for these 4-dimensional representations, our specialization results enable us to verify that Hida's results relating (a) the 2-variable Rankin-Selberg $p$-adic $L$-function to the 3-variable Rankin Selberg $p$-adic $L$-function and (b) the 1-variable Rankin-Selberg $p$-adic $L$-function to the 2-variable Rankin-Selberg $p$-adic $L$-function, are consistent with our control theorems for Selmer groups.

Email : pbharath [AT] math [dot] upenn [.] edu

Office : 4N53 David Rittenhouse Labs