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 (last updated June, 2018) Acknowledgement: Archives of the Mathematisches Forschungsinstitut Oberwolfach

## Papers and Preprints

1. On Selmer groups and factoring p-adic L-functions,
accepted for publication in International Mathematics Research Notices
(Journal) (Arxiv) (Video)

2. Show/Hide Abstract

Samit Dasgupta has proved a formula factoring a certain restriction of a $3$-variable Rankin-Selberg $p$-adic $L$-function as a product of a $2$-variable $p$-adic $L$-function related 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 $p$-adic $L$-function is associated), the $3$-dimensional representation (to which the $2$-variable $p$-adic $L$-function is associated) and the $1$-dimensional representation (to which the Kubota-Leopoldt $p$-adic $L$-function is associated). Under certain additional hypotheses, we indicate how one can use work of Urban to deduce main conjectures for the $3$-dimensional representation and the $4$-dimensional representation. One key technical input to our methods is studying the behavior of Selmer groups under specialization.

3. Height one specializations of Selmer groups,
accepted for publication in Annales de l'Institut Fourier
(Arxiv) (Sage Worksheet)

4. Show/Hide Abstract

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 the Hida family $G$, where $f$ is a classical specialization of $F$ with weight $k \geq 2$. We prove 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 its 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, Hida has constructed $1$-variable, $2$-variable and $3$-variable Rankin-Selberg $p$-adic $L$-functions. Our specialization results enable us to verify that Hida's results relating (a) the $2$-variable $p$-adic $L$-function to the $3$-variable $p$-adic $L$-function and (b) the $1$-variable $p$-adic $L$-function to the $2$-variable $p$-adic $L$-function and our control theorems for Selmer groups are completely consistent with the main conjectures.

5. On free resolutions of Iwasawa modules, (Preprint)
with Alexandra Nichifor

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. This application involves a careful study of the Dieudonné determinant. 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.

6. Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction, (Preprint)
with Antonio Lei
Sage Code 1, Sage Code 2

Show/Hide Abstract

A recent result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's $2$-variable $p$-adic $L$-functions) and algebraic objects (two everywhere unramified'' Iwasawa modules) involving codimension two cycles in a $2$-variable Iwasawa algebra. We prove an analogous result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$, defined over $\mathbb{Q}$, with good supersingular reduction at $p$. On the analytic side, we consider eight out of the ten pairs of $2$-variable $p$-adic $L$-functions in this setup (four of the five $2$-variable $p$-adic $L$-functions have been constructed by Loeffler and the fifth $2$-variable $p$-adic $L$-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the $\mathbb{Z}_p^2$-extension of $K$. We also provide numerical evidence, using algorithms of Pollack, towards a pseudo-nullity conjecture of Coates-Sujatha.

7. Codimension two cycles in Iwasawa theory and tensor product of Hida families (in preparation), with Antonio Lei

## Contact

Email : pbharath [AT] math [dot] upenn [.] edu
Office : 4N53 David Rittenhouse Labs