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Anwesh Ray

Publications and source records attributed to Anwesh Ray.

At least 37 records · Page 2Linked to original sources
arXiv2025

Deformations of reducible Galois representations with large Selmer $p$-rank

Let $p\geq 5$ be a prime number. In this paper, we construct Galois representations associated with modular forms for which the dimension of the $p$-torsion in the Bloch-Kato Selmer group can be made arbitrarily large. Our result extends similar results known for small primes, such as Matsuno's work on Tate-Shafarevich groups of elliptic curves. Extending the technique of Hamblen and Ramakrishna, we lift residually reducible Galois representations to modular representations for which the associated Greenberg Selmer groups are minimally generated by a large number of elements over the Iwasawa algebra. We deduce that there is an isogenous lattice for which the Bloch-Kato Selmer group has large $p$-rank.

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arXiv2025

Selmer stability in families of congruent Galois representations

In this article I study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$. Motivated by analogies with Goldfeld's conjecture on ranks in quadratic twist families of elliptic curves, I investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg's local conditions under congruences of residual Galois representations. Let $X$ be a positive real number. Fix a residual representation $\bar{\rho}$ and a corresponding modular form $f$ of weight $2$ and optimal level. I count the number of level-raising modular forms $g$ of weight $2$ that are congruent to $f$ modulo $p$, with level $N_g\leq X$, such that the $p$-rank of the Selmer groups of $g$ equals that of $f$. Under some mild assumptions on $\bar{\rho}$, I prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$, for an explicit constant $\alpha > 0$. The main result is a partial generalization of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.

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arXiv2025

Selmer stability for elliptic curves in Galois $\ell$-extensions

We study the behavior of Selmer groups of an elliptic curve $E/\mathbb{Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime $\ell \geq 5$, a finite group $G$ with $\#G = \ell^n$, and an elliptic curve $E/\mathbb{Q}$ with $Sel_\ell(E/\mathbb{Q}) = 0$ and surjective mod-$\ell$ Galois representation. We show that there exist infinitely many Galois extensions $F/\mathbb{Q}$ with Galois group $Gal(F/\mathbb{Q}) \simeq G$ for which the $\ell$-Selmer group $Sel_\ell(E/F)$ also vanishes. We obtain an asymptotic lower bound for the number $M(G, E; X)$ of such fields $F$ with absolute discriminant $|\Delta_F|\leq X$, proving that there is an explicit constant $\delta>0$ such that $M(G, E; X) \gg X^{\frac{1}{\ell^{n-1}(\ell - 1)}} (\log X)^{\delta - 1}$. The asymptotic for $M(G, E; X)$ matches the conjectural count for all $G$-extensions $F/\mathbb{Q}$ for which $|\Delta_F|\leq X$, up to a power of $\log X$. This demonstrates that Selmer stability is not a rare phenomenon.

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arXiv2025

Iwasawa theory and the representations of finite groups

In this note, I develop a representation-theoretic refinement of the Iwasawa theory of finite Cayley graphs. Building on analogies between graph zeta functions and number-theoretic L-functions, I study $\mathbb{Z}_\ell$-towers of Cayley graphs and the asymptotic growth of their Jacobians. My main result establishes that the Iwasawa polynomial associated to such a tower admits a canonical factorization indexed by the irreducible representations of the underlying group. This leads to the definition of representation-theoretic Iwasawa polynomials, whose properties are studied.

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arXiv2025

On the Iwasawa invariants of Artin representations

We study Iwasawa invariants associated to Selmer groups of Artin representations, and criteria for the vanishing of the associated algebraic Iwasawa invariants. The conditions obtained can be used to study natural distribution questions in this context.

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arXiv2025

An analogue of Kida's formula in graph theory

Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the graphs in the pullback are connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the assumption that the Iwasawa $μ$-invariant of the tower above $X$ vanishes, we prove a formula relating the Iwasawa $λ$-invariant of the $\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $λ$-invariant of the pullback. This formula is analogous to Kida's formula in classical Iwasawa theory. We present an application to the study of structural properties of certain noncommutative pro-$\ell$ towers of graphs, based on an analogy with classical results of Cuoco on the growth of Iwasawa invariants in $\mathbb{Z}_\ell^2$-extensions of number fields. Our investigations are illustrated by explicit examples.

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arXiv2025

Diophantine approximation and the subspace theorem

Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results to higher dimensions, with profound implications to Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth's theorem and Schlickewei's refinement of the subspace theorem, with an emphasis on proofs. The arguments presented are classical and approachable for readers with a background in algebraic number theory, serving as a streamlined, yet condensed reference for these fundamental results.

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arXiv2024

Rank stability of elliptic curves in certain non-abelian extensions

Let $E_{/\mathbb{Q}}$ be an elliptic curve with rank $E(\mathbb{Q})=0$. Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K/\mathbb{Q}$ with rank $E(K) = 0$. In this paper, we show that there exist infinitely many extensions $L/K$ such that $L/\mathbb{Q}$ is Galois with $\operatorname{Gal}(L/\mathbb{Q}) \simeq \operatorname{Gal}(K/\mathbb{Q}) \ltimes \mathbb{Z}/p^n\mathbb{Z}$, and rank $E(L)=0$. This is an extension of earlier results on rank stability of elliptic curves in cyclic extensions of prime power order to a non-abelian setting. We also obtain an asymptotic lower bound for the number of such extensions, ordered by their absolute discriminant.

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arXiv2024

Iwasawa theory and ranks of elliptic curves in quadratic twist families

We study the distribution of ranks of elliptic curves in quadratic twist families using Iwasawa-theoretic methods, contributing to the understanding of Goldfeld's conjecture. Given an elliptic curve $ E/\mathbb{Q} $ with good ordinary reduction at $ 2 $ and $ λ_2(E/\mathbb{Q}) = 0 $, we use Matsuno's Kida-type formula to construct quadratic twists $ E^{(d)} $ such that $ λ_2(E^{(d)}/\mathbb{Q}) $ remains unchanged or increases by $ 2 $. When the root number of $E^{(d)}$ is $-1$ and the Tate-Shafarevich group $Sha(E^{(d)}/\mathbb{Q})[2^\infty] $ is finite, this yields quadratic twists with Mordell--Weil rank $ 1 $. These results support the conjectural expectation that, on average, half of the quadratic twists in a family have rank $ 0 $ and half have rank $ 1 $. In the cases we consider we obtain asymptotic lower bounds for the number of twists by squarefree numbers $d\leq X$ which match with the conjectured value up to an explicit power of $\log X$. They complement recent groundbreaking results of Smith on Goldfeld's conjecture.

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arXiv2024

Galois representations over function fields that are ramified at one prime

Let $\mathbb{F}_q$ be the finite field with $q$ elements, $F:=\mathbb{F}_q(T)$ and $F^{\operatorname{sep}}$ a separable closure of $F$. Set $A$ to denote the polynomial ring $\mathbb{F}_q[T]$. Let $\mathfrak{p}$ be a non-zero prime ideal of $A$, and $\mathscr{O}$ be the completion of $A$ at $\mathfrak{p}$. Given any integer $r\geq 2$, I construct a Galois representation $ρ:\operatorname{Gal}(F^{\operatorname{sep}}/F)\rightarrow \operatorname{GL}_r(\mathscr{O})$ which is unramified at all non-zero primes $\mathfrak{l}\neq \mathfrak{p}$ of $A$, and whose image is a finite index subgroup of $\operatorname{GL}_r(\mathscr{O})$. Moreover, if the degree of $\mathfrak{p}$ is $1$, then $ρ$ is also unramified at $\infty$.

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arXiv2024

Rank distribution in cubic twist families of elliptic curves

Let $a$ be an integer which is not of the form $n^2$ or $-3 n^2$ for $n\in \mathbb{Z}$. Let $E_a$ be the elliptic curve with rational $3$-isogeny defined by $E_a:y^2=x^3+a$, and $K:=\mathbb{Q}(μ_3)$. Assume that the $3$-Selmer group of $E_a$ over $K$ vanishes. It is shown that there is an explicit infinite set of cubefree integers $m$ such that the $3$-Selmer groups over $K$ of $E_{m^2 a}$ and $E_{m^4 a}$ both vanish. In particular, the ranks of these cubic twists are seen to be $0$ over $K$. Our results are proven by studying stability properties of $3$-Selmer groups in cyclic cubic extensions of $K$, via local and global Galois cohomological techniques.

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arXiv2024

Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$

Let $\ell\geq 5$ be a prime number and $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $GL_2(\mathbb{F}_\ell)$ and absolute discriminant bounded above by $X$ is asymptotically at least $\frac{X^{\frac{\ell}{12(\ell-1)\# GL_2(\mathbb{F}_\ell)}}}{\log X}$. We also obtain a similar result for the number of surjective homomorphisms $ρ:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{F}_\ell)$ ordered by the prime to $\ell$ part of the Artin conductor of $ρ$.

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arXiv2024

On the Iwasawa theory of Cayley graphs

This paper explores Iwasawa theory from a graph theoretic perspective, focusing on the algebraic and combinatorial properties of Cayley graphs. Using representation theory, we analyze Iwasawa-theoretic invariants within $\mathbb{Z}_\ell$-towers of Cayley graphs, revealing connections between graph theory, number theory, and group theory. Key results include the factorization of associated Iwasawa polynomials and the decomposition of $μ$- and $λ$-invariants. Additionally, we apply these insights to complete graphs, establishing conditions under which these invariants vanish.

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arXiv2024

Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension

Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation $n=x^3+y^3$, where $x$ and $y$ belong to the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, consider the case when $n$ is not a sum of rational cubes. Then, we prove that $n$ cannot be a sum of two cubes in certain large families of prime cyclic extensions of $\mathbb{Q}$.

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arXiv2024

A Heuristic approach to the Iwasawa theory of elliptic curves

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime such that $E$ has good ordinary reduction at $p$ and the Galois representation on $E[p]$ is irreducible. Then Greenberg's $\mu=0$ conjecture predicts that the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ is cofinitely generated as a $\mathbb{Z}_p$-module. In this article we study this conjecture from a statistical perspective. We extend the heuristics of Poonen and Rains to obtain further evidence for Greenberg's conjecture. The key idea is that the vanishing of the $\mu$-invariant can be detected by the intersection $M_1\cap M_2$ of two Iwasawa modules $M_1, M_2$ with additional properties in a given inner product space. The heuristic is based on showing that there is a probability measure on the space of pairs $(M_1, M_2)$ respect to which the event that $M_1\cap M_2$ is finite happens with probability $1$.

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arXiv2024

Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions

We explore distribution questions for rational maps on the projective line $\mathbb{P}^1$ over $\mathbb{Q}$ within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems for rational maps $\phi$ of fixed degree $d \geq 2$ with prescribed reduction properties. Our main result establishes that the set of rational maps with minimal resultant has positive density. Additionally, for degree 2 rational maps, we perform explicit computations demonstrating that over $32.7\%$ possess a squarefree, and hence minimal, resultant.

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arXiv2024

Galois representations are surjective for almost all Drinfeld modules

This article advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let $F$ be the rational function field over a finite field. I establish that for Drinfeld modules of rank $r \geq 2$, the $T$-adic Galois representation: $\widehat{\rho}_{\phi, T}: Gal(F^{sep}/F) \rightarrow GL_r(\mathbb{F}_q[[T]])$ is surjective for a density $1$ set of such modules. The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's uniformization theory and sieve methods.

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