arXiv subjects

Anwesh Ray

Publications and source records attributed to Anwesh Ray.

At least 19 recordsLinked to original sources

arXiv2026

Adelic Galois representations are maximal for almost all Drinfeld modules

Let $r\geq 3$ be an integer, let $q$ be an odd prime power, and set $A=\mathbb{F}_q[T]$ and $F=\mathbb{F}_q(T)$. We study the Galois representations associated with rank $r$ Drinfeld modules of the form $ϕ_T=T+g_1τ+\cdots+g_rτ^r$, where $g_i\in A$ and $g_r\neq 0$. Ordering the coefficient tuples $\vec{g}=(g_1,\ldots,g_r)$ by their maximum degree, we prove that, for a density-one set of tuples, the $λ$-adic Galois representation is surjective for every prime $λ$ of $A$, and the adelic Galois image contains $\mathrm{SL}_r(\widehat A)$. Moreover, for every tuple in this set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides $q-1$. These results extend Zywina's theorem in rank $2$ to arbitrary ranks $r\geq 3$. As a further consequence, we establish the existence of rank $r$ Drinfeld modules with surjective adelic Galois representations for every $r\geq 3$, extending previous existence results which were known for prime values of $r$.

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arXiv2026

Counting Nilpotent Galois extensions with prescribed genus number

Let $G$ be a nontrivial finite nilpotent group. In this paper, we study the distribution of genus numbers of Galois extensions of $\Q$ with Galois group isomorphic to $G$, ordered by absolute discriminant. We show that the proportion of fields with any prescribed genus number tends to zero, and obtain a quantitative bound for this proportion. Moreover, if every element of order $\ell$ in $G$ is central, where $\ell$ is the smallest prime divisor of $|G|$, we determine the orders of magnitude of the average genus number and its higher moments.

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arXiv2026

On the distribution of shapes of octic Kummer extensions

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers under linear operations generated by rotations, reflections, and positive scalar dilations. It may be viewed as a point in the space of shapes $\mathscr{S}_{n-1} = \mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$. In this paper, we study the distribution of shapes of octic Kummer extensions $L=\mathbb{Q}(i,\sqrt[4]{m})$, where $m\in\mathbb{Z}[i]$ is fourth-power-free. We parametrize these shapes by explicit invariants known as shape parameters and establish an asymptotic formula for their joint distribution ordered by absolute discriminant. The limiting distribution is given by an explicit measure that factors as the product of a continuous measure and a discrete measure arising from local arithmetic conditions.

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arXiv2026

Shapes of tame abelian number fields

The shape of a number field $K$ of degree $n\geq2$ is the similarity class of the lattice obtained by projecting $\mathscr{O}_K$ orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-$(n-1)$ lattices, $\mathscr{S}_{n-1}:=\mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$, where $\mathrm{GO}_{n-1}(\mathbb{R})$ is the group of orthogonal similitudes. Let $G$ be a fixed nontrivial finite abelian group, and let $K/\mathbb{Q}$ vary over tamely ramified Galois extensions with Galois group isomorphic to $G$. We study the variation of the shapes of such number fields in $\mathscr S_{n-1}$, where $n=|G|$.

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arXiv2026

On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape

We study the distribution of $2$-torsion in class groups and narrow class groups of cubic fields and cubic orders subject to prescribed shape conditions. The \emph{shape} of a cubic order in a number field is a natural geometric invariant taking values in the modular surface $\mathbb{H}/\operatorname{GL}_2(\mathbb{Z})$. Fix a subset $W$ of the modular surface with positive hyperbolic measure and boundary of measure zero. Refining the methods of Bhargava and Varma, we prove that among cubic fields with shape in $W$, the average size of the $2$-torsion subgroup of the class group is $5/4$ for totally real fields and $3/2$ for complex fields, while the average size of the $2$-torsion subgroup of the narrow class group for totally real cubic fields is $2$. We also obtain analogous results for cubic orders satisfying prescribed local conditions at all primes.

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arXiv2026

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 Bhargava et al., we prove unconditionally that a positive proportion of imaginary quadratic fields meet our criterion when $p=3$. For $p=11,13,31,37$, the analogous conclusions obtained from the rank-zero twist families of Kriz--Li are conditional on the vanishing of the $p$-primary Tate--Shafarevich groups for a positive relative proportion of those twists.

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arXiv2026

Counting number fields with symplectic Galois group

Let $n\geq 1$ and let $\ell$ be an odd prime. Let $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ or $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$, and fix a faithful transitive permutation representation $π:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permutation representation is $π$. For $σ\in S_d$, write $\operatorname{ind}(σ)$ for its permutation index, namely $\operatorname{ind}(σ) = d-\#\{\text{orbits of $σ$ on $\{1,\ldots,d\}$}\}$. If $τ$ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most $X$ is bounded below by a constant multiple of $X^{1/(2n\,\operatorname{ind}(π(τ)))}$. For the natural vector and projective actions, the exponents we obtain are asymptotically $1/(4n)$ of those predicted by the weak form of Malle's conjecture as $\ell\to\infty$. The fields are constructed from the mod-$\ell$ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.

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arXiv2026

Iwasawa theory of class field towers

Let $p$ be an odd prime and let $K_\infty/K$ be a $\mathbb{Z}_p$-extension. For each finite layer $K_n$, let $G_n$ be the Galois group of its maximal unramified pro-$p$ extension. We study the augmentation and Zassenhaus filtrations of the groups $G_n$ as $n$ tends to infinity. Assume that the classical Iwasawa $μ$-invariant is positive. If $d_n$ denotes the minimal number of generators of $G_n$, then there is an integer $δ>0$ such that $d_n=δp^n+O(1)$, while the minimal number of relations is $O(p^n)$. For each fixed $m\geq2$, as $n\to\infty$, we show that the $m$-th augmentation quotient has the same leading asymptotic as that of a free pro-$p$ group on $d_n$ generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.

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arXiv2026

Local-global principles for visibility of lattice points on parameterized curves

We develop a local-global theory for visibility of lattice points on families of parameterized curves. We introduce a notion of $p$-adic visibility and ask whether a lattice point is globally visible precisely when it is visible at every prime. We prove that this local-global principle holds for a broad class of families whose parametrizations are homogeneous with respect to positive weights. When the points on these curves fill the entire positive integer lattice, we compute the local and global densities of visible points and show that the global density is the product of the local densities. We then consider polynomial families of the form $y=qP(x)$, with $q\in\mathbb{Q}_{>0}$, and show that visibility can be detected prime by prime exactly when $P$ is a monomial. For non-monomial polynomials the local-global principle can fail, but the set of points where it fails has density zero; for separable polynomials we also obtain a quantitative bound for the number of non-visible points, improving the previously known bound. We further consider families whose lattice points lie on a proper lower-dimensional algebraic subset of the ambient space and show that their visibility densities can behave differently from those of the full lattice. Finally, we extend the theory from visibility from the origin to visibility from one lattice point to another and show that the corresponding local-global principle continues to hold for weighted homogeneous families.

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arXiv2026

Counting number fields whose Galois group is a wreath product of symmetric groups

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 $|Δ_L|:=|Norm_{K/\mathbb{Q}} (Δ_{L/K})|$. 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.

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arXiv2026

Massey products and the Iwasawa theory of fine Selmer groups

A central conjecture of Coates and Sujatha predicts that the fine Selmer group of any $p$-adic Galois representation is cotorsion over the relevant Iwasawa algebra with vanishing $μ$-invariant, generalizing Iwasawa's original conjecture for class groups. In this article, we recast this conjecture in terms of higher Galois cohomological operations called Massey products, stated purely in terms of the residual representation. This characterization implies that if the $μ$-invariant vanishes for a given $\mathbb{Z}_p$-extension, then it also vanishes for all $\mathbb{Z}_p$-extensions the Greenberg neighbourhood of radius $1/p$. Furthermore, we establish that the unobstructedness of ordinary deformation rings over $\mathbb{Z}_p$-extensions is equivalent to the vanishing of the $μ$-invariant of the fine Selmer group attached to the adjoint representation. For ordinary deformation rings attached to $2$-dimensional automorphic Galois representations, we establish a close connection between the Noetherian property and the vanishing of the $μ$-invariant.

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arXiv2026

Galois representations ramified at one prime via relative deformation theory

Let $p$ be an odd prime and let $\mathbf{G}/\mathbb Z$ be a split connected reductive group with $\operatorname{dim} Z(\mathbf{G})\leq1$. Assume that a split maximal torus of $\mathbf{G}$ admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of $\mathrm{Lie}(\mathbf{G}^{\mathrm{der}})$ modulo $p$. We construct infinitely many continuous representations $ρ:G_{\{p\}}\longrightarrow\mathbf{G}(\mathbb{Z}_p)$ which are unramified at every finite prime different from $p$ and whose images contain a principal congruence subgroup of $\mathbf G^{\mathrm{der}}(\mathbb{Z}_p)$. When the center has dimension one, the representations may be chosen to have open image in $\mathbf{G}(\mathbb{Z}_p)$. The construction continues the author's earlier work on $\mathrm{GL}_n$-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For $\mathbf{G}=\mathrm{GL}_n$ all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd $p$ and every $n>1$, infinitely many representations $G_{\{p\}}\to\mathrm{GL}_n(\mathbb{Z}_p)$ with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction $p\geq7$ in that construction.

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arXiv2026

Cyclotomic norm congruences for zeta values and modular $L$-values

We study congruences between special values of zeta functions and modular $L$-functions in cyclotomic towers. The underlying mechanism is an integral norm congruence for finite-level Iwasawa-theoretic elements, which becomes stronger as one ascends the tower. In the $\mathrm{GL}_1$ setting, this gives congruences for Dedekind zeta values over totally real fields, with applications to higher $K$-groups, generalized Bernoulli numbers, and Euler-Poincaré characteristics. In the $\mathrm{GL}_2$ setting, we use Mazur-Tate elements and their distribution relations to control the primitive twisted special values appearing at each cyclotomic level. Via Artin formalism, these give congruences for the corresponding base-change $L$-values over successive cyclotomic fields.

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arXiv2026

On the number of subrings of $\mathbb{Z}^n$ of prime power index

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 $α=(α_1, \dots, α_{n-1})$ of $e$ into $n-1$ positive integers, let $g_α(p)$ denote the number of irreducible subrings of $\mathbb{Z}^n$ for which the associated upper triangular matrix in Hermite normal form has diagonal $(p^{α_1}, \dots, p^{α_{n-1}},1)$. Via combinatorial analysis, the computation of $f_n(p^e)$ reduces to the computation of $g_α(p)$ 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 $g_α(p)$ for certain infinite families of compositions $α$, for which we are able to obtain general estimates using techniques from the geometry of numbers.

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arXiv2026

On the distribution of shapes of pure quartic number fields

The shape of a number field is a subtle arithmetic invariant arising from the geometry of numbers. It is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. For a number field of degree $n$, the shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $GL_{n-1}(\mathbb{Z}) \backslash GL_{n-1}(\mathbb{R}) / GO_{n-1}(\mathbb{R})$. In this paper, we investigate the distribution of shapes in the family of pure quartic fields $K_m = \mathbb{Q}(\sqrt[4]{m})$. We prove that the shape of $K_m$ lies on one of ten explicitly described torus orbits in $\mathscr{S}_3$, determined by the sign and residue class of $m \bmod 32$. It is shown that the shape on a given torus orbit is completely determined by two parameters, one of which varies continuously, while the other takes values in a discrete set. As a result, the distribution of shapes in this family is governed by a product of a continuous and a discrete measure. Our results shed new light on a question posed by Manjul Bhargava and Piper H concerning the distribution of shapes in families of non-generic number fields of fixed degree. Notably, the limiting distribution in our case does not arise as the restriction of the natural measure on $\mathscr{S}_3$ induced by Haar measure on $GL_3(\mathbb{R})$.

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arXiv2026

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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arXiv2026

Greenberg's $μ=0$ conjecture for lisse sheaves over global function fields

Let $K$ be a global function field of characteristic $p>0$ and $\ell\neq p$ be a prime number. We study Selmer groups over a $\mathbb{Z}_\ell$-extension $K_\infty/K$. For a lisse $\mathbb Z_\ell$-sheaf we prove that the Pontryagin dual of the associated Selmer group is a finitely generated torsion module over the Iwasawa algebra and has $μ$-invariant equal to zero. This gives a positive-characteristic, prime to $p$, analogue of Greenberg's $μ=0$ conjecture. Our result applies in particular to abelian varieties, fine Selmer groups, and adjoint representations. We also prove an analogue of the weak Leopoldt conjecture in this context over $K_\infty$, and deduce that the framed deformation ring of a residual representation is a formal power series ring. The same conclusion holds for the unframed deformation ring if the residual representation has no non-scalar endomorphisms.

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arXiv2026

An analogue of Kida's formula for Mazur-Tate elements

We prove an analogue of Kida's formula for the Iwasawa invariants of the Mazur-Tate elements attached to elliptic curves over $\mathbb{Q}$. Let $p$ be an odd prime and let $L/K$ be a Galois extension of abelian number fields with $p$-power Galois group. For an elliptic curve $E/\mathbb{Q}$, we study the Mazur-Tate elements over the finite layers of the cyclotomic $\mathbb{Z}_p$-extensions of $K$ and $L$. We show that the vanishing of the $μ$-invariant is preserved in the extension: if the level-$n$ Mazur-Tate element over $K$ has $μ= 0$, then the corresponding element over $L$ also has $μ= 0$. Moreover, the associated $λ$-invariants satisfy an explicit transition formula. This parallels the work of Hachimori-Matsuno on Selmer groups and of Matsuno on $p$-adic $L$-functions. As an application, we obtain an analogue of Kida's formula for the analytic Iwasawa invariants associated to Pollack's signed $p$-adic $L$-functions. Since our results apply to elliptic curves with any reduction type at $p$ under mild hypotheses, including those with additive reduction, we also obtain a Kida-type formula for the $p$-adic $L$-functions constructed by Delbourgo for elliptic curves with unstable additive reduction. In particular, because Mazur-Tate elements approximate $p$-adic $L$-functions in the limit, our results unify all previously known cases of Kida's formula for analytic Iwasawa invariants.

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