I am a Postdoctoral Fellow at the National Center for Theoretical Sciences, Taiwan. Previously, I was a Hans Rademacher instructor in the Mathematics Department at the University of Pennsylvania. I completed my PhD in the Mathematics Department at the University of Washington where my thesis advisor was Ralph Greenberg. My research interests in algebraic number theory are motivated by questions in Iwasawa theory and Hida theory. My CV (last updated June, 2019) Conference: $L$-values and Iwasawa theory photo credit: Archives of the Mathematisches Forschungsinstitut Oberwolfach

## Papers and Preprints

1. On Selmer groups and factoring p-adic L-functions,
International Mathematics Research Notices (2018), Issue 24, December 2018, Pages 7483-7554
(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,
Annales de l'Institut Fourier, Volume 69 (2019) no. 1, p. 303-334
(Journal) (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,
with Alexandra Nichifor
Doc. Math. 24, 609-662 (2019)
(Journal) (arXiv)

Show/Hide Abstract

Let $\Lambda$ (isomorphic to $\mathbb{Z}_p[[T]]$) denote the usual Iwasawa algebra and $G$ denote the Galois group of a finite Galois extension $L/K$ of totally real fields. When the non-primitive Iwasawa module over the cyclotomic $\mathbb{Z}_p$-extension has a free resolution of length one over the group ring $\Lambda[G]$, we prove that 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 integrality result involves a careful study of the Dieudonn\'e determinant. Using a cohomolgoical criterion of Greenberg, we also deduce the precise conditions under which the non-primitive Iwasawa module has a free resolution of length one. As one application of the last result, 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,
with Antonio Lei,
Forum of Mathematics, Sigma (2019), 7, E25.
(Journal) (arXiv) (Sage Code 1) (Sage Code 2)

Show/Hide Abstract

A 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 a 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 pairs of $2$-variable $p$-adic $L$-functions in this setup (four of the $2$-variable $p$-adic $L$-functions have been constructed by Loeffler and a 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
with Antonio Lei,
accepted for publication in Mathematische Annalen
(arXiv) (Journal)

Show/Hide Abstract

The purpose of this paper is to build on results in higher codimension Iwasawa theory. The setting of our results involves Galois representations arising as cyclotomic twist deformations associated to (i) the tensor product of two cuspidal Hida families $F$ and $G$, and (ii) the tensor product of three cuspidal Hida families $F$, $G$ and $H$. On the analytic side, we consider (i) a pair of $3$-variable Rankin-Selberg $p$-adic $L$-functions constructed by Hida and (ii) a balanced $4$-variable $p$-adic $L$-function (whose construction is forthcoming in a work of Hsieh and Yamana) and an unbalanced $4$-variable $p$-adic $L$-function (whose existence is currently conjectural). In each of these setups, when the two $p$-adic $L$-functions generate a height two ideal in the corresponding deformation ring, we use codimension two cycles of that ring to relate them to a pair of pseudo-null modules.

## Teaching

### At University of Pennsylvania

1. Fall 2018 (access restricted)
2. Fall 2016 (access restricted)

1. Fall 2018 (access restricted)
2. Spring 2017 (access restricted)

1. Spring 2018 (access restricted)

1. Fall 2016 (access restricted)

## Contact

Email : pbharath [AT] math [dot] upenn [DOT] edu
National Center for Theoretical Sciences, Taiwan