arXiv subjects
Publications and source records attributed to Anwesh Ray.
Let $\ell$ be a prime number, $F$ be a global function field of characteristic $\ell$. Assume that there is a prime $P_\infty$ of degree $1$. Let $\mathcal{O}_F$ be the ring of functions in $F$ with no poles outside of $\{P_\infty\}$. We study solutions to Catalan's equation $X^m-Y^n=1$ over $\mathcal{O}_F$ and show that under certain additional conditions, there are no non-constant solutions which lie in $\mathcal{O}_F$, when $m,n>1$.
We study arithmetic statistics for Iwasawa invariants for fine Selmer groups associated to elliptic curves.
The Iwasawa $μ$-invariant of the Selmer group of a residually reducible Galois representation arising from a Hecke eigencuspform is studied. Furthermore, certain Iwasawa-invariants refining the $μ$-invariant are defined and analyzed. As an application, we show that given any reducible mod-$p$ Galois representation $\barρ$ and any choice of integer $N\geq 1$, there is a modular Galois representation lifting $\barρ$ whose associated Selmer group has $μ$-invariant $\geq N$. This is a refinement of Serre's conjecture in the residually reducible case.
Let $K$ be a number field and $k\geq 2$ be an integer. Let $(n_1,n_2, \dots, n_k)$ be a vector with entries $n_i\in \mathbb{Z}_{\geq 2}$. Given a number field extension $L/K$, we denote by $\widetilde{L}$ the Galois closure of $L$ over $K$. We prove asymptotic lower bounds for the number of number field extensions $L/K$ with $[L:K]=\prod_{i=1}^k n_i$, such that $Gal(\widetilde{L}/K)$ is isomorphic to the iterated wreath product of symmetric groups $S_{n_1}\wr S_{n_2}\wr \dots \wr S_{n_k}$. Here, the number fields $L$ are ordered according to discriminant $|\Delta_L|:=|Norm_{K/\mathbb{Q}} (\Delta_{L/K})|$. The results in this paper are motivated by Malle's conjecture. When $n_1=n_2=\dots =n_k$, these wreath products arise naturally in the study of arboreal Galois representations associated to rational functions over $K$. We prove our results by developing Galois theoretic techniques that have their origins in the study of dynamical systems.
Given an odd prime number $p$ and an imaginary quadratic field $K$, we establish a relationship between the $p$-rank of the class group of $K$, and the classical $λ$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $K$. Exploiting this relationship, we prove statistical results for the distribution of $λ$-invariants for imaginary quadratic fields ordered according to their discriminant. Some of our results are conditional since they rely on the original Cohen--Lenstra heuristics for the distribution of the $p$-parts of class groups of imaginary quadratic fields. Some results are unconditional results ad are obtained by leveraging theorems of Byeon, Craig and others.
Let $p\geq 5$ be a prime, and $\mathfrak{p}$ a prime of $\bar{\mathbb{Q}}$ above $p$. Let $g_1$ and $g_2$ be $\mathfrak{p}$-ordinary, $\mathfrak{p}$-distinguished and $p$-stabilized cuspidal newforms of nebentype characters $ε_1, ε_2$ respectively, and weight $k\geq 2$, whose associated newforms have level prime to $p$. Assume that the residual representations at $\mathfrak{p}$ associated to $g_1$ and $g_2$ are absolutely irreducible and isomorphic. Then, the imprimitive $p$-adic L-functions associated with the symmetric square representations are shown to exhibit a congruence modulo $\mathfrak{p}$. Furthermore, the analytic and algebraic Iwasawa invariants associated to these representations of the $g_i$ are shown to be related. Along the way, we give a complete proof of the integrality of the $\mathfrak{p}$-adic L-function, normalized with Hida's canonical period. This fills a gap in the literature, since, despite the result being widely accepted, no complete proof seems to ever have been written down. On the algebraic side, we establish the corresponding congruence for Greenberg's Selmer groups, and verify that the Iwasawa main conjectures for the twisted symmetric square representations for $g_1$ and $g_2$ are compatible with the congruences.
Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$ which are unramified outside $\{p, \infty\}$ with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod-$p^t$ reduction map $SL_n(\mathbb{Z}_p)\rightarrow SL_n(\mathbb{Z}/p^t\mathbb{Z})$, where $t:=8(n^2-n)\left(3+\lfloor log_p(2^n+1)\rfloor\right)+8$. The results are proven via a purely Galois theoretic lifting construction. When $p\equiv 1\mod{4}$, our results are conditional since in this case, we assume a very weak version of Vandiver's conjecture.
In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of prime degree. We also study growth of the $p$-primary Selmer group and the Shafarevich--Tate group in cyclic degree-$p$ extensions and improve upon previously known results in this direction.
Let $\ell$ be a prime number and $q$ be a power of $\ell$. Given an odd prime number $p$ and an imaginary quadratic extension $F$ of the rational function field $\mathbb{F}_q(T)$, let $λ_p(F)$ denote the Iwasawa $λ$-invariant of the constant $\mathbb{Z}_p$-extension of $F$. We show that for any number $r>0$ and all large enough values of $q\not\equiv 1\mod{p}$, there is a positive proportion of imaginary quadratic fields $F/\mathbb{F}_q(T)$ with the property that $λ_p(F)\geq r$. The main result is proved as a consequence of recent unconditional theorems of Ellenberg-Venkatesh-Westerland on the distribution of class groups of imaginary quadratic function fields.
Let $p\geq 5$ be a prime. We construct modular Galois representations for which the $\mathbb{Z}_p$-corank of the $p$-primary Selmer group (i.e., $λ$-invariant) over the cyclotomic $\mathbb{Z}_p$-extension is large. More precisely, for any natural number $n$, one constructs a modular Galois representation such that the associated $λ$-invariant is $\geq n$. The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form $f_1$ satisfying suitable conditions, one constructs a congruent modular form $f_2$ for which the $λ$-invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakruddin-Khare-Patrikis, which extends previous work of Ramakrishna. The results are subject to certain additional hypotheses, and are illustrated by explicit examples.
We study the Iwasawa theory of the fine Selmer group associated to certain Galois representations. The vanishing of the $μ$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We outline conditions under which the $μ=0$ conjecture is shown to hold for various Galois representations of interest.
We study the distribution of arithmetic invariants associated to Alexander polynomials for certain infinite families of links. The families of links we consider arise from braids on a fixed number of strings. We explore analogies with number theory and the distribution of class groups in various families of number fields, setting out new directions in arithmetic topology and arithmetic statistics.
Let $n$ and $k$ be positive integers, and $f_n(k)$ (resp. $g_n(k)$) be the number of unital subrings (resp. unital irreducible subrings) of $\mathbb{Z}^n$ of index $k$. The numbers $f_n(k)$ are coefficients of certain zeta functions of natural interest. The function $k\mapsto f_n(k)$ is multiplicative, and the study of the numbers $f_n(k)$ reduces to computing the values at prime powers $k=p^e$. Given a composition $\alpha=(\alpha_1, \dots, \alpha_{n-1})$ of $e$ into $n-1$ positive integers, let $g_\alpha(p)$ denote the number of irreducible subrings of $\mathbb{Z}^n$ for which the associated upper triangular matrix in Hermite normal form has diagonal $(p^{\alpha_1}, \dots, p^{\alpha_{n-1}},1)$. Via combinatorial analysis, the computation of $f_n(p^e)$ reduces to the computation of $g_\alpha(p)$ for all compositions of $i$ into $j$ parts, where $i\leq e$ and $j\leq n-1$. We extend results of Liu and Atanasov-Kaplan-Krakoff-Menzel, who explicitly compute $f_n(p^e)$ for $e\leq 8$. The case $e=9$ proves to be significantly more involved. We evaluate $f_n(e^9)$ explicitly in terms of a polynomial in n and p up to a single term which is conjecturally a polynomial. Our results provide further evidence for a conjecture, which states that for any fixed pair $(n,e)$, the function $p\mapsto f_n(p^e)$ is a polynomial in $p$. A conjecture of Bhargava on the asymptotics for $f_n(k)$ as a function of $k$ motivates the study of the asymptotics for $g_\alpha(p)$ for certain infinite families of compositions $\alpha$, for which we are able to obtain general estimates using techniques from the geometry of numbers.
Let $X$ be a proper smooth toric variety over a perfectoid field of prime residue characteristic $p$. We study the perfectoid space $\mathcal{X}^{perf}$ which covers $X$ constructed by Scholze, showing that $\text{Pic}(\mathcal{X}^{perf})$ is canonically isomorphic to $\text{Pic}(X)[p^{-1}]$. We also compute the cohomology of line bundles on $\mathcal{X}^{perf}$ and establish analogs of Demazure and Batyrev-Borisov vanishing. This generalizes the first author's analogous results for "projectivoid space".
Given a prime $p$ and $q$ a power of $p$, we study the statistics of $p$-ranks of Artin--Schreier covers of given genus defined over $\mathbb{F}_q$, in the large $q$-limit. We refer to this problem as the geometric problem. We also study an arithmetic variation of this problem, and consider Artin--Schreier covers defined over $\mathbb{F}_p$, letting $p$ go to infinity. Distribution of $p$-ranks has been previously studied for Artin--Schreier covers over a fixed finite field as the genus is allowed to go to infinity. The method requires that we count isomorphism classes of covers that are unramified at $\infty$.
Let $p$ be an odd prime and $F_\infty$ be a $\mathbb{Z}_p$-extension of a number field $F$. Given an elliptic curve $E$ over $F$, we study the structure of the fine Selmer group over $F_\infty$. It is shown that under certain conditions, the fine Selmer group is a cofinitely generated module over $\mathbb{Z}_p$ and furthermore, we obtain an upper bound for its corank (i.e., the $λ$-invariant), in terms of various local and global invariants.
Let $\ell$ and $p$ be (not necessarily distinct) prime numbers and $F$ be a global function field of characteristic $\ell$ with field of constants $κ$. Assume that there exists a prime $P_\infty$ of $F$ which has degree $1$, and let $\mathcal{O}_F$ be the subring of $F$ consisting of functions with no poles away from $P_\infty$. Let $f(X)$ be a polynomial in $X$ with coefficients in $κ$. We study solutions to diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal{O}_F$, and in particular, show that if $m$ and $f(X)$ satisfy additional conditions, then there are no non-constant solutions. The results obtained apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb{Z}_{p}$-extensions of $F$ known as constant $\mathbb{Z}_p$-extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \dots, T_r]$, where $K$ is any field of characteristic $\ell$, showing that the only solutions must lie in $K$. We apply our methods to study solutions of diophantine equations of the form $Y^n=\sum_{i=1}^d (X+ir)^m$, where $m,n, d\geq 2$ are integers.
Let $E$ be an elliptic curve with positive rank over a number field $K$ and let $p$ be an odd prime number. Let $K_{cyc}$ be the cyclotomic $\mathbb{Z}_p$-extension of $K$ and $K_n$ denote its $n$-th layer. The Mordell--Weil rank of $E$ is said to be constant in the cyclotomic tower of $K$ if for all $n$, the rank of $E(K_n)$ is equal to the rank of $E(K)$. We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in the above sense. We then indicate the potential applications to Hilbert's tenth problem for number rings.