arXiv subjects

Aprameyo Pal

Publications and source records attributed to Aprameyo Pal.

9 recordsLinked to original sources

arXiv2026

The size of $2$-Selmer groups for the $\frac{\pi}{3}$-congruent number problem

Our main objective in this paper is to study the average rank of the $2$-Selmer group of the elliptic curve associated with the $\frac{\pi}{3}$-congruent number problem. Following Heath-Brown's strategy, we could find an asymptotic formula for the size of the relaxed $2$-Selmer groups, which has several consequences towards the average of $2$-Selmer ranks and $\frac{\pi}{3}$-congruent number problem. Indeed, we could find an unconditional positive density of $2$-Selmer rank being $1$ or $3$, among the positive square-free integers $n\equiv 13\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$ and an unconditional positive density of $2$-Selmer rank being $0$ or $2$, among the positive square-free integers $n\equiv 5\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$.

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arXiv2025

Perfectoid Spaces in Multivariate $p$-adic Hodge Theory

Perfectoid spaces have become a crucial tool in $p$-adic geometry, serving as a bridge between adic spaces in characteristic $0$ and those in characteristic $p$. In this article, we develop a systematic way to study the structure of perfectoid spaces within the setting of multivariate $p$-adic Hodge theory over a variant of the rings introduced in \cite{Bri}.

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arXiv2025

A $p$-Converse theorem for Real Quadratic Fields

Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let $\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\mathbb{Z}} \hspace{0.01mm} \hspace{1mm} E(F) = 1$ and $\#\Big(\underline{III}(E/F)_ {p^\infty}\Big) < \infty$, then $ord_{s=1} \ L(E/F,s) = 1$. Further, we give an application to a $p$-converse theorem over $\mathbb{Q}$.

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arXiv2025

(Algebraic) $ p $-adic Artin formalism of twisted triple product Galois representations over real quadratic fields

In this article, we investigate factorization problems for twisted triple product Galois representations over real quadratic fields, arising from families of Hilbert cusp forms. Specifically, we address the factorization in two distinct settings determined by the order of vanishing of associated $L$-unctions at their central critical values-namely, the rank (1,1) and rank (0,2) cases. Our results generalize the algebraic factorization framework developed by B\"{u}y\"{u}kboduk et al. in higher rank scenario to the setting of real quadratic fields. Notably, our work yields the first known factorization result in the higher rank (0,2) case, marking a significant advancement in the study of triple product motives over totally real fields.

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arXiv2024

On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points

Our main objective in this paper (which is expository for the most part) is to study the necessary steps to prove a factorization formula for a certain triple product $p$-adic $L$-function guided by the Artin formalism. The key ingredients are: a) the explicit reciprocity laws governing the relationship of diagonal cycles and generalized Heegner cycles to $p$-adic $L$-functions; b) a careful comparison of Chow--Heegner points and twisted Heegner points in Hida families, via formulae of Gross--Zagier type.

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arXiv2022

Sufficient conditions for a problem of Polya

Let $\alpha$ be a non-zero algebraic number. Let $K$ be the Galois closure of $\mathbb{Q}(\alpha)$ with Galois group $G$ and $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. In this article, among the other results, we prove the following. If $f\in \bar{\mathbb{Q}}[G]$ is a non-zero element of the group ring $\bar{\mathbb{Q}}[G]$ and $\alpha$ is a given algebraic number such that $f(\alpha^n)$ is a non-zero algebraic integer for infinitely many natural numbers $n$, then $\alpha$ is an algebraic integer. This result generalizes the result of Polya [11], Corvaja and Zannier [2] and Philippon and Rath [9]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [4], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al., [6] which are applications of the Schmidt subspace theorem.

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arXiv2020

Automorphic $\mathrm{SL}_2$-periods and the subconvexity problem for $\mathrm{GL}_2 \times \mathrm{GL}_3$

We prove a new (conditional) result towards the subconvexity problem for certain automorphic $L$-functions for $\mathrm{GL}_2 \times \mathrm{GL}_3$. This follows from the computation of new $\mathrm{SL}_2$-period integrals associated with newforms $f$ and $g$ of even weight and odd squarefree level. The same computations also lead to a central value formula for degree $6$ complex $L$-series of the form $L(f \otimes \mathrm{Ad}(g), s)$, extending previous work in arXiv:1804.02352.

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arXiv2019

Pullbacks of Saito--Kurokawa lifts and a central value formula

We prove an explicit central value formula for a family of complex $L$-series of degree $6$ for $\mathrm{GL}_2 \times \mathrm{GL}_3$ which arise as factors of certain Garret--Rankin triple product $L$-series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito--Kurokawa lifts, and as an application, we prove Deligne's conjecture stating the algebraicity of the central values of the considered $L$-series up to the relevant periods.

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arXiv2019

Cohomology and overconvergence for representations of powers of Galois groups

We show that the Galois cohomology groups of $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ can be computed via the generalization of Herr's complex to multivariable $(φ,Γ)$-modules. Using Tate duality and a pairing for multivariable $(φ,Γ)$-modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ are overconvergent and, moreover, passing to overconvergent multivariable $(φ,Γ)$-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

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