arXiv subjects

Bharathwaj Palvannan

Publications and source records attributed to Bharathwaj Palvannan.

11 recordsLinked to original sources

arXiv2026

On the congruence ideal associated to $p$-adic families of Yoshida lifts

We study congruences involving $p$-adic families of Hecke eigensystems of Yoshida lifts associated with two Hida families (say $\mathcal{F},\mathcal{G}$) of elliptic cusp forms. With appropriate hypotheses, we show that if a Hida family of genus two Siegel cusp forms admits a Yoshida lift at an appropriately chosen classical specialization, then all classical specializations are Yoshida lifts. Moreover, we prove that the characteristic ideal of the non-primitive Selmer group of (a self-dual twist of) the Rankin--Selberg convolution of $\mathcal{F}$ and $\mathcal{G}$ is divisible by the congruence ideal of the Yoshida lift associated with $\mathcal{F}$ and $\mathcal{G}$. Under an additional assumption inspired by pseudo-nullity conjectures in higher codimension Iwasawa theory, we establish the pseudo-cyclicity of the dual of the primitive Selmer group over the cyclotomic $\mathbb{Z}_p$-extension.

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arXiv2026

Modularity theorems for Eisenstein congruences in prime-power level

Let $p, N \geq 5$ be primes such that $N \equiv 1 \bmod p$. We prove modularity theorems at levels $N$ and $N^2$, showing that suitable Eisenstein localizations of the weight-$2$ $p$-adic Hecke algebra at these levels are isomorphic to certain natural quotients of a universal pseudodeformation ring. This universal ring parametrizes pseudorepresentations that are residually Eisenstein, unramified outside $Np$ and finite-flat at $p$, and satisfy appropriate conditions at $N$ depending on the level.

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arXiv2026

A minimal modularity lifting theorem for Siegel modular forms

We prove a minimal modularity lifting theorem (in the spirit of Genestier--Tilouine and Pilloni) in the setting of Siegel modular forms of genus two when the residual representation arises from a stable Yoshida lift, that is, an automorphic induction of a nearly ordinary Hilbert modular eigencuspform over a real quadratic field. As applications of the underlying $R=\mathbb{T}$ theorem, we establish the freeness of a universal minimal ordinary Galois deformation ring over an Iwasawa algebra in two variables along with the uniqueness of Hida families passing through classical $p$-ordinary Siegel modular eigenforms with very regular weights.

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arXiv2026

On local characterizations of Hida families of Siegel modular forms

We provide new local characterizations of Hida families of Siegel modular forms with genus two arising from stable Yoshida lifts, that is, automorphic inductions of nearly ordinary Hilbert modular eigenforms over real quadratic fields. Our characterizations involve (i) density of de Rham at $p$ specializations at the singular weights $(k,2)$ and (ii) local decomposability at $p$ of the associated $Λ$-adic Galois representation. These are analogous to the characterizations of Hida families of CM modular forms provided by Ghate--Vatsal. Our approach is similar to that of Castella--Wang-Erickson who provided an alternate strategy to reproving Ghate--Vatsal's main results by applying Ribet's method when an anti-cyclotomic class group is assumed to be pseudo-null and cyclic as a $Λ$-module. Along these lines, one key input to our methods involves an assumption of pseudo-nullity of Selmer groups that are defined by imposing stricter conditions at $p$ than those imposed for the usual Greenberg Selmer groups appearing in the Asai main conjectures over real quadratic fields.

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arXiv2026

A new perspective on the rank of Mazur's Eisenstein Hecke algebra

Let $N, p \geq 5$ be primes such that $N \equiv 1 \bmod p$. We study the rank $r$ of the Hecke algebra that parametrizes modular forms of weight 2 and level $N$ that are Eisenstein modulo $p$. When $r$ is $2$ or $3$, we prove that $r-1$ equals the order of vanishing of the mod-$p$ reduction of a zeta element that interpolates Dirichlet $L$-values at $-1$, thereby recovering results of Merel and Lecouturier. This equality can fail in some cases when $r \geq 4$, and we provide a heuristic explanation of this failure. Our approach handles all of these cases uniformly by studying the analogous Hecke algebra in level $N^2$. When exactly one of $r-1$ or the order of vanishing equals $3$, we provide precise information about Galois orbits of cuspidal newforms in level $N^2$ that are Eisenstein modulo $p$.

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arXiv2025

An ergodic approach towards an equidistribution result of Ferrero--Washington

An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic $μ$-invariant for Kubota--Leopoldt $p$-adic $L$-functions is an equidistribution result which they established using the Weyl criterion. The purpose of our manuscript is to provide an alternative proof by adopting a dynamical approach. A key ingredient to our methods is studying an ergodic skew-product map on $\mathbb{Z}_p \times [0,1]$, which is then suitably identified as a factor of the $2$-sided Bernoulli shift on the sample space $\{0,1,2,\cdots,p-1\}^{\mathbb{Z}}$.

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arXiv2020

Codimension two cycles in Iwasawa theory and tensor product of Hida families

The purpose of this paper is to build on results in {\it{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 (due to 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.

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arXiv2019

On free resolutions of Iwasawa modules

Let $Λ$ (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 $Λ[G]$, 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 $Λ$-order. This integrality result involves a careful study of the Dieudonné 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 $Λ$ of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over $Λ$, associated to two non-primitive classical Iwasawa~modules.

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arXiv2019

Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction

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.

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arXiv2018

Height one specializations of Selmer groups

We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to four 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 one variable, two variable and three variable Rankin-Selberg $p$-adic $L$-functions. Our specialization results enable us to verify that Hida's results relating (a) the two variable $p$-adic $L$-function to the three variable $p$-adic $L$-function and (b) the one variable $p$-adic $L$-function to the two variable $p$-adic $L$-function and our control theorems for Selmer groups are completely consistent with the main conjectures.

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arXiv2017

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

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.

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