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Ravitheja Vangala

Publications and source records attributed to Ravitheja Vangala.

5 recordsLinked to original sources
arXiv2026

Reductions Of Crystalline Representations Of Fractional Slope $<p-1$

Let $p$ be an odd prime and let $V_{k,a_p}$ be the two-dimensional crystalline representation of the Galois group of ${\mathbb Q}_p$ of weight $k \geq 2$ and parameter $a_p \in \bar{\mathbb{Q}}_p$. We study the semi-simplification $\bar{V}_{k,a_p}$ of the mod $p$ reduction of $V_{k,a_p}$ when the slope (valuation of $a_p$) is a positive fraction $< p-1$ using the mod $p$ local Langlands correspondence. We describe the $\textit{exact shape}$ of $\bar{V}_{k,a_p}$ for all such slopes and all (sufficiently large, depending on the slope) weights $k$, as long as certain Jordan-H\"older factors of dimension $p-1$ do not intervene in the computation (when $k$ is odd), though we also provide some criteria which further determine the shape of $\bar{V}_{k,a_p}$ in some of these exceptional cases. To keep this paper a reasonable length, we assume that for certain bad congruence classes of $k$ mod $p$, the slope is less than the representative - taken in the range $[1,p-1]$ - of the congruence class of $k-2$ mod $(p-1)$, which is generically the case if the slope is small. Finally, a folklore conjecture predicts that the reduction $\bar{V}_{k,a_p}$ is $\textit{irreducible}$ for fractional slopes if $k$ is even. We deduce this conjecture for all fractional slopes $< p-2$ and all (sufficiently large, even) weights $k$ under the aforementioned slope assumption.

math.NT↗
arXiv2022

Iwasawa theory for Rankin-Selberg convolution at an Eisenstein prime

Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and $ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l < k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty} $-Selmer group of the Rankin-Selberg product of $f$ and $h$ under the assumption that $ p $ is an Eisenstein prime for $ h $ i.e. the residual Galois representation of $ h $ at $ p $ is reducible. We show that the $ p $-adic $ L $-function and the characteristic ideal of the $p^\infty$-Selmer group of the Rankin-Selberg product of $f, h$ generate the same ideal modulo $ p $ in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$ holds mod $p$. As an application to our results, we explicitly describe a few examples where the above congruence holds.

math.NT↗
arXiv2019

$p$-adic Asai $L$-functions attached to Bianchi cusp forms

We establish a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and construct distributions interpolating these L-values. Using the method of abstract Kummer congruences, we then outline the main steps needed to show that these distributions come from a measure.

math.NT↗
arXiv2019

The Monomial Lattice in Modular Symmetric Power Representations

Let $p$ be a prime. We study the structure of and the inclusion relations among the terms in the monomial lattice in the modular symmetric power representations of $\mathrm{GL}_2(\mathbb{F}_p)$. We also determine the structure of certain related quotients of the symmetric power representations which arise when studying the reductions of local Galois representations of slope at most $p$. In particular, we show that these quotients are periodic and depend only on the congruence class modulo $p(p-1)$. Many of our results are stated in terms of the sizes of various sums of digits in base $p$-expansions and in terms of the vanishing or non-vanishing of certain binomial coefficients modulo $p$.

math.RT↗