Anwesh Ray
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View article: Mazur's growth number conjecture and congruences
Mazur's growth number conjecture and congruences Open
Motivated by the work of Greenberg-Vatsal and Emerton-Pollack-Weston, I investigate the extent to which Mazur's conjecture on the growth of Selmer ranks in $\mathbb{Z}_p$-extensions of an imaginary quadratic field persists under $p$-congru…
View article: The second moment of cubic Dirichlet L-functions over function fields
The second moment of cubic Dirichlet L-functions over function fields Open
In this article, we study the second moment of cubic Dirichlet L-functions at the central point $s=1/2$ over the rational function field $\mathbb{F}_q(T)$, where $q$ is a power of an odd prime satisfying $q \equiv 2 \pmod{3}$. Our result e…
View article: Selmer stability in families of congruent Galois representations
Selmer stability in families of congruent Galois representations Open
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…
View article: Deformations of reducible Galois representations with large Selmer $p$-rank
Deformations of reducible Galois representations with large Selmer $p$-rank Open
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 s…
View article: Selmer stability for elliptic curves in Galois $\ell$-extensions
Selmer stability for elliptic curves in Galois $\ell$-extensions Open
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 $E/\mathbb{Q}$ wi…
View article: Iwasawa theory and the representations of finite groups
Iwasawa theory and the representations of finite groups Open
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 gr…
View article: Diophantine approximation and the subspace theorem
Diophantine approximation and the subspace theorem Open
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 result…
View article: Iwasawa theory and ranks of elliptic curves in quadratic twist families
Iwasawa theory and ranks of elliptic curves in quadratic twist families Open
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 redu…
View article: CLASS GROUP STATISTICS FOR TORSION FIELDS GENERATED BY ELLIPTIC CURVES
CLASS GROUP STATISTICS FOR TORSION FIELDS GENERATED BY ELLIPTIC CURVES Open
For a prime p and a rational elliptic curve $E_{/\mathbb {Q}}$ , set $K=\mathbb {Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname {ker}\{E\xrightarrow {p} E\}$ . The class group $\operatorname {Cl}_K$ is a module ove…
View article: Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension
Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension Open
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…
View article: A Heuristic approach to the Iwasawa theory of elliptic curves
A Heuristic approach to the Iwasawa theory of elliptic curves Open
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 $μ=0$ conjecture predicts that the Selmer group of $E…
View article: Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions
Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions Open
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 f…
View article: Galois representations are surjective for almost all Drinfeld modules
Galois representations are surjective for almost all Drinfeld modules Open
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 esta…
View article: A positive density of elliptic curves are diophantine stable in certain Galois extensions
A positive density of elliptic curves are diophantine stable in certain Galois extensions Open
Let $p \in \{3, 5\}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $ E $ defined over $ \mathbb{Q} $, ordered by height, that are diophantine stable in $ L $.
View article: Galois representations over function fields that are ramified at one prime
Galois representations over function fields that are ramified at one prime Open
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 o…
View article: Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields
Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields Open
Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are i…
View article: On the Iwasawa theory of Cayley graphs
On the Iwasawa theory of Cayley graphs Open
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 $\mathbb{Z}_\ell$-t…
View article: Statistics for Iwasawa invariants of elliptic curves, $\rm{III}$
Statistics for Iwasawa invariants of elliptic curves, $\rm{III}$ Open
Given a prime $p\geq 5$, a conjecture of Greenberg predicts that the $μ$-invariant of the $p$-primary Selmer group should vanish for most elliptic curves with good ordinary reduction at $p$. In support of this conjecture, I show that the $…
View article: Rank distribution in cubic twist families of elliptic curves
Rank distribution in cubic twist families of elliptic curves Open
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_…
View article: Asymptotic growth patterns for class field towers
Asymptotic growth patterns for class field towers Open
Let p be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a \mathbb{Z}_{p} -tower. These Galois groups can be considered as non-commutative a…
View article: On Malle's conjecture for the product of symmetric and nilpotent groups
On Malle's conjecture for the product of symmetric and nilpotent groups Open
Let $G$ be a finite nilpotent group and $n\in \{3,4, 5\}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{|G|}\subset S_{n|G|}$, where $G$ embeds into the second factor of $S_n\times S_{|G|}$ via the regular representation. Over any…
View article: Rank stability of elliptic curves in certain non-abelian extensions
Rank stability of elliptic curves in certain non-abelian extensions Open
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 man…
View article: The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules
The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules Open
Let $\mathbb{F}_q$ be the finite field with $q\geq 5$ elements, $A:=\mathbb{F}_q[T]$ and $F:=\mathbb{F}_q(T)$. Assume that $q$ is odd and take $|\cdot|$ to be the absolute value at $\infty$ that is normalized by $|T|=q$. Given a pair $w=(g…
View article: Iwasawa theory of fine Selmer groups associated to Drinfeld modules
Iwasawa theory of fine Selmer groups associated to Drinfeld modules Open
Let $q$ be a prime power and $F=\mathbb{F}_q(T)$ be the rational function field over $\mathbb{F}_q$, the field with $q$ elements. Let $ϕ$ be a Drinfeld module over $F$ and $\mathfrak{p}$ be a non-zero prime ideal of $A:=\mathbb{F}_q[T]$. O…
View article: Upper bounds for the number of number fields with prescribed Galois group
Upper bounds for the number of number fields with prescribed Galois group Open
Let $n$ be a positive integer and $G$ be a transitive permutation subgroup of $S_n$. Given a number field $K$ with $[K:\mathbb{Q}]=n$, we let $\widetilde{K}$ be its Galois closure over $\mathbb{Q}$ and refer to $Gal(\widetilde{K}/\mathbb{Q…
View article: ON THE DISTRIBUTION OF IWASAWA INVARIANTS ASSOCIATED TO MULTIGRAPHS
ON THE DISTRIBUTION OF IWASAWA INVARIANTS ASSOCIATED TO MULTIGRAPHS Open
Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $ -towers of multigraphs. In this context, growth patterns are realized by certain…
View article: On the Iwasawa invariants of Artin representations
On the Iwasawa invariants of Artin representations Open
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…
View article: Asymptotic growth patterns for class field towers
Asymptotic growth patterns for class field towers Open
Let $p$ be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a $\mathbb{Z}_p$-tower. These Galois groups can be considered as non-commutative …