S. Chidambaram

activity

20192026

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12 papers · 1 filter

math.NT2026

Computing residual representations of abelian threefolds with imaginary multiplication

Shiva Chidambaram, Pip Goodman

Let $M$ be an imaginary quadratic field. Let $A$ be a polarised abelian threefold defined over $M$ with geometric endomorphism algebra isomorphic to $M$ . We study residual Galois r…

math.NT2026

Point counts of abelian varieties over finite fields determining their zeta function

Shiva Chidambaram, Timo Keller

Let $A$ be an abelian variety of dimension $g$ over a finite field $F_{q}$ . We show that if $q$ is sufficiently large relative to $g$ , the $g$ point counts $\#A(\mathbf{F}_{…

math.NT2026

On the Visibility category of the Shafarevich--Tate group

Barinder S. Banwait, Jerson Caro, Shiva Chidambaram

Given an elliptic curve $E$ over $\Q$ and a nontrivial element $σ$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the \textbf{Visualization category} $\V(E; σ)$ of abelian…

math.NT2025

Computing the mod-3 Galois image of a principally polarized abelian surface over the rationals

Shiva Chidambaram

A lot of work has gone into computing images of Galois representations coming from elliptic curves. This article presents an algorithm to determine the image of the mod- $3$ Galois…

math.NT2023

Fekete polynomials of principal Dirichlet characters

Shiva Chidambaram, Ján Mináč, Tung T. Nguyen +1

Fekete polynomials associated to quadratic Dirichlet characters have interesting arithmetic properties, and have been studied in many works. In this paper, we study a seemingly sim…

math.NT2023

Computing isogeny classes of typical principally polarized abelian surfaces over the rationals

Raymond van Bommel, Shiva Chidambaram, Edgar Costa +1

We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $Q$ with geometric endomorphism ring equal to $Z$ , compute…

math.NT2022

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

Nikola Adžaga, Shiva Chidambaram, Timo Keller +1

We complete the computation of all $Q$ -rational points on all the $64$ maximal Atkin-Lehner quotients $X_{0} (N)^{*}$ such that the quotient is hyperelliptic. To achieve this,…

math.NT2021

Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

Nikola Adžaga, Vishal Arul, Lea Beneish +4

We use the method of quadratic Chabauty on the quotients $X_{0}^{+} (N)$ of modular curves $X_{0} (N)$ by their Fricke involutions to provably compute all the rational points of these curv…

math.NT2020

Mod- $p$ Galois representations not arising from abelian varieties

Shiva Chidambaram

It is known that any Galois representation $ρ : G_{Q} \to GL (2, F_{p})$ with determinant equal to the mod- $p$ cyclotomic character, arises from the $…

math.NT2020

Rationality of twists of the Siegel modular variety of genus $2$ and level $3$

Frank Calegari, Shiva Chidambaram

Let $ρ : G_{Q} \to GSp_{4} (F_{3})$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equiva…

math.NT2020

Abelian surfaces with fixed $3$ -torsion

Frank Calegari, Shiva Chidambaram, David P. Roberts

Given a genus two curve $X : y^{2} = x^{5} + a x^{3} + b x^{2} + c x + d$ , we give an explicit parametrization of all other such curves $Y$ with a specified symplectic isomorphism on three-…

math.NT2019

Some modular abelian surfaces

Frank Calegari, Shiva Chidambaram, Alexandru Ghitza

We use the main theorem of Boxer-Calegari-Gee-Pilloni (arXiv:1812.09269) to give explicit examples of modular abelian surfaces $A$ over $Q$ without extra endomorhpisms suc…

Computing residual representations of abelian threefolds with imaginary multiplication

Point counts of abelian varieties over finite fields determining their zeta function

On the Visibility category of the Shafarevich--Tate group

Computing the mod-3 Galois image of a principally polarized abelian surface over the rationals

Fekete polynomials of principal Dirichlet characters

Computing isogeny classes of typical principally polarized abelian surfaces over the rationals

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

Mod-p Galois representations not arising from abelian varieties

Rationality of twists of the Siegel modular variety of genus 2 and level 3

Abelian surfaces with fixed 3-torsion

Some modular abelian surfaces

Mod- $p$ Galois representations not arising from abelian varieties

Rationality of twists of the Siegel modular variety of genus $2$ and level $3$

Abelian surfaces with fixed $3$ -torsion