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Anwesh Ray

Publications and source records attributed to Anwesh Ray.

At least 55 records · Page 3Linked to original sources
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.

math.NT↗
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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arXiv2024

Class group statistics for torsion fields generated by elliptic curves

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 over $\operatorname{Gal}(K/\mathbb{Q})$. Given a fixed odd prime number $p$, we study the average non-vanishing of certain Galois stable quotients of the mod-$p$ class group $\operatorname{Cl}_K/p\operatorname{Cl}_K$. Here, $E$ varies over rational elliptic curves, ordered according to \emph{height}. Our results are conditional and rely on predictions made by Delaunay and Poonen-Rains for the statistical variation of the $p$-primary parts of Tate-Shafarevich groups of elliptic curves. We also prove results in the case when the elliptic curve $E_{/\mathbb{Q}}$ is fixed and the prime $p$ is allowed to vary.

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arXiv2024

An analogue of Kida's formula for elliptic curves with additive reduction

We study the Iwasawa theory of $p$-primary Selmer groups of elliptic curves $E$ over a number field $K$. Assume that $E$ has additive reduction at the primes of $K$ above $p$. In this context, we prove that the Iwasawa invariants satisfy an analogue of the Riemann--Hurwitz formula. This generalizes a result of Hachimori and Matsuno. We apply our results to study rank stability questions for elliptic curves in prime cyclic extensions of $\mathbb{Q}$. These extensions are ordered by their absolute discriminant and we prove an asymptotic lower bound for the density of extensions in which the Iwasawa invariants as well as the rank of the elliptic curve is stable.

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arXiv2024

Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields

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 identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Kriz--Li and Bhargava et al., we demonstrate that for primes $ p = 3, 11, 13, 31, 37 $, a positive proportion of imaginary quadratic fields meet our criteria.

math.NT↗
arXiv2024

Iwasawa theory of fine Selmer groups associated to Drinfeld modules

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 $A:=\mathbb{F}_q[T]$. Over the constant $\mathbb{Z}_p$-extension of $F$, we introduce the fine Selmer group associated to the $\mathfrak{p}$-primary torsion of $ϕ$. We show that it is a cofinitely generated module over $A_{\mathfrak{p}}$. This proves an analogue of Iwasawa's $μ=0$ conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article.

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arXiv2024

Statistics for Anticyclotomic Iwasawa Invariants of Elliptic Curves

We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves, considered over anticyclotomic $\mathbb{Z}_p$-extensions in both the definite and indefinite settings. The results in this paper lie at the intersection of arithmetic statistics and Iwasawa theory.

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arXiv2024

Statistics for Iwasawa invariants of elliptic curves, $\rm{III}$

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 $5$-primary Iwasawa $μ$- and $λ$-invariants simultaneously vanish for an explicit positive density of elliptic curves $E_{/\mathbb{Q}}$. The elliptic curves in question have good ordinary reduction at $5$, and are ordered by their height. The results are proven by leveraging work of Bhargava and Shankar on the distribution of $5$-Selmer groups of elliptic curves defined over $\mathbb{Q}$.

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arXiv2024

Hilbert's tenth problem in Anticyclotomic towers of number fields

Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits in $K$. Let $E_1$ and $E_2$ be elliptic curves over $K$ such that the $Gal(\bar{K}/K)$-modules $E_1[p]$ and $E_2[p]$ are isomorphic. We show that under certain explicit additional conditions on $E_1$ and $E_2$, the anticyclotomic $\mathbb{Z}_p$-extension $K_{anti}$ of $K$ is integrally diophantine over $K$. When such conditions are satisfied, we deduce new cases of Hilbert's tenth problem. In greater detail, the conditions imply that Hilbert's tenth problem is unsolvable for all number fields that are contained in $K_{anti}$. We illustrate our results by constructing an explicit example for $p=3$ and $K=\mathbb{Q}(\sqrt{-5})$.

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arXiv2024

Arithmetic statistics for Galois deformation rings

Given an elliptic curve $E$ defined over the rational numbers and a prime $p$ at which $E$ has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the $p$-torsion group $E[p]$. For a fixed elliptic curve without complex multiplication, it is shown that these deformation rings are unobstructed for all but finitely many primes. For a fixed prime $p$ and varying elliptic curve $E$, we relate the problem to the question of how often $p$ does not divide the modular degree. Heuristics due to M.Watkins based on those of Cohen and Lenstra indicate that this proportion should be $\prod_{i\geq 1} \left(1-\frac{1}{p^i}\right)\approx 1-\frac{1}{p}-\frac{1}{p^2}$. This heuristic is supported by computations which indicate that most elliptic curves (satisfying further conditions) have smooth deformation rings at a given prime $p\geq 5$, and this proportion comes close to $100\%$ as $p$ gets larger.

math.NT↗
arXiv2024

Constructing Galois representations with prescribed Iwasawa $λ$-invariant

Let $p\geq 5$ be a prime number. We consider the Iwasawa $λ$-invariants associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. Let $g$ be a $p$-ordinary cuspidal newform of weight $2$ and trivial nebentype. We assume that the $μ$-invariant of $g$ vanishes, and that the image of the residual representation associated to $g$ is suitably large. We show that for any number greater $n$ greater than or equal to the $λ$-invariant of $g$, there are infinitely many newforms $f$ that are $p$-congruent to $g$, with $λ$-invariant equal to $n$. We also prove quantitative results regarding the levels of such modular forms with prescribed $λ$-invariant.

math.NT↗
arXiv2024

Asymptotic growth patterns for class field towers

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 analogues of ray class groups. For certain $\mathbb{Z}_p$-extensions in which a given prime above $p$ is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for $p$-primary class numbers of the number fields in a $\mathbb{Z}_p$-tower.

math.NT↗
arXiv2023

The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules

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 $w=(g_1, g_2)\in A^2$ with $g_2\neq 0$, consider the associated Drinfeld module $ϕ^w: A\rightarrow A\{τ\}$ of rank $2$ defined by $ϕ_T^w=T+g_1τ+g_2τ^2$. Fix integers $c_1, c_2\geq 1$ and define $|w|:=max\{|g_1|^{\frac{1}{c_1}}, |g_2|^{\frac{1}{c_2}}\}$. I show that when ordered by height, there is a positive density of pairs $w=(g_1, g_2)$, such that the $T$-adic Galois representation attached to $ϕ^w$ is surjective.

math.NT↗
arXiv2023

Statistics for Iwasawa Invariants of elliptic curves, $\rm{II}$

We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain unconditional lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.

math.NT↗
arXiv2023

Upper bounds for the number of number fields with prescribed Galois group

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 $Gal(\widetilde{K}/\mathbb{Q})$ as its Galois group. We may identify this Galois group with a transitive subgroup of $S_n$. Given a real number $X>0$, we set $N_{n}(X;G)$ to be the number of such number fields $K$ for which the absolute discriminant is bounded above by $X$, and for which $Gal(\widetilde{K}/\mathbb{Q})$ is isomorphic to $G$ as a permutation subgroup of $S_n$. We prove an asymptotic upper bound for $N_n(X;G)$ as $X\rightarrow\infty$. This result is conditional and based upon the non-vanishing of certain polynomial determinants in $n$-variables. We expect that these determinants are non-vanishing for many groups, and demonstrate through some examples how they may be computed.

math.NT↗
arXiv2023

On the distribution of Iwasawa invariants associated to multigraphs

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 analogues of Iwasawa invariants, which depend on the prime $\ell$ and the abelian $\ell$-tower of multigraphs. We formulate and study statistical questions about the behaviour of the Iwasawa $μ$ and $λ$ invariants.

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