Papers

Let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, respectively, and consider \[ E_k:\qquad y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1). \] These elliptic curves arise naturally from the regular Diophantine triples \[ \{F_{2k},F_{2k+2},F_{2k+4}\}. \] For odd $k$, we exhibit the rational point \[ Q_k=\left( -\frac{F_{k-1}}{L_kF_{k+1}F_{k+2}}, \frac{F_{2k+1}}{L_kF_{k+1}F_{k+2}} \right). \] For every odd $k\geq 3$, this point is independent of the standard point $P_k=(0,1)$; in particular, $\operatorname{rank}E_k(\mathbb{Q})\geq 2$. Moreover, if $k\geq 3$ is odd and $\operatorname{rank}E_k(\mathbb{Q})=2$, then all integer points on $E_k$ are exactly the points arising from the two known solutions of the Hoggatt-Bergum extension problem. By parametrizing the two conics $L^2-5F^2=\pm4$ and applying an injective specialization criterion, we also show that the corresponding one-parameter elliptic families have generic ranks $2$ in the odd case and $1$ in the even case. Finally, we discuss computational data and propose the heuristic rank distribution $1/4,1/2,1/4$ for ranks $1,2,3$, respectively, with density zero for rank at least $4$.

We investigate automorphic forms for congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$ that are Iwahori at $p$. In particular, we study the cuspidal cohomology of Iwahori congruence subgroups $\mathcal{I}(3, p)$, which are comprised of matrices in $\mathrm{SL}(3, \mathbb{Z})$ that are upper triangular modulo $p$. In order to work with $\mathcal{I}(3, p)$, we generalize key results from Ash, Grayson, and Green [J.\ Number Theory 19 (1984), pp.\ 412-436]. For levels $\mathcal{I}(3, p)$ with $p \leq 227$, we found three levels with nonzero cuspidal classes and were able to compute the action of Hecke operators at two of these levels. These are the first examples of non-essentially-self-dual automorphic representations of trivial cohomological weight that are Steinberg at $p$ appearing at Iwahori level.

For quadratic postcritically finite polynomials defined over a number field $K$, we establish lower bounds for the Call-Silverman height of wandering algebraic integers lying in cyclotomic extensions of $K$ and provide partial results for the nonintegral case. Our proof combines potential theory and algebraic number theory with equidistribution techniques.

For integers $h,k\geq 1$, let $hA$ be the $h$-fold sumset of $A$ and put $\mathcal{R}(h,k)=\{|hA|:A\subset\mathbb{Z}, |A|=k\}$. Previously, the fixed-cardinality exponent law was known only for $k\leq 3$; every fixed $k\geq 4$ remained open. We settle the problem in full by determining the complete fixed-cardinality exponent law: $|\mathcal{R}(h,k)|=\begin{cases}1,&k\leq 2,\\ h,&k=3,\\ h^{k-1+o_k(1)},&k\geq 4\end{cases}$. Here $o_k(1)\to 0$ as $h\to\infty$ with $k$ fixed. More sharply, for fixed $k\geq 4$, an interval of length $\Theta_k(h^{k-1})$ contains at least $h^{k-1-o_k(1)}$ attainable values. At $k=4$ we prove $|\mathcal{R}(h,4)|=\Theta(h^3)$ with positive lower density in its ambient interval, disproving Nathanson's proposed $o(h^3)$ and $O(h^2)$ bounds. One bounded addition-table geometry drives these results, coupling Hilbert-energy amplification to optimal finite-observation compression. Every ordered real $k$-set ($k\geq 2$) has an integer model in $[0,O_k(h^{k-2})]$ preserving every sum equality and strict comparison through degree $h$; the exponent $k-2$ is sharp. The universal label-realization length is therefore $\Theta_k(h^{k-2})$, one power sharper than Nathanson's $O_k(h^{k-1})$ bound. For $h\geq 2$ and $k\geq 3$, minimum active rank equals realization-frequency codimension, exponent-shape codimension, and sampling-rarity exponent; a full-exponent family has maximal-rank witnesses with Cohen-Macaulay toric coordinate rings. At rank zero, for $h\geq 2$, it proves the conjectural OEIS A227589 formula $\binom{h+2}{2}+\mathbf{1}_{\{2\nmid h\}}$ for the least normalized diameter of a four-point $B_h$-set. It also gives exact fixed-$(h,k)$ popularity laws for $k$-subsets of $\{1,\ldots,q\}$ as $q\to\infty$, resolving Nathanson's Problems 9 and 10.

Hilbert's Tenth Problem over the integers was solved negatively by Y. Matiyasevich in 1970. In this paper, we prove that there is no algorithm to determine for any polynomial equation $P(z_1,\ldots,z_{10})=0$ (with integer coefficients and ten unknowns) whether it has integer solutions. This improves the previous 11 unknowns theorem.

Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3. After reducing the partition to primitive common-period data, Fourier cancellation produces a finite inverse-sine system. We prove that no such system can exist when every density is below 1/3. The proof combines a divisor-concentration identity with uniform analytic estimates and three exact finite verifications, all carried out with integer or rational arithmetic. The component supplied by the density bound has mean spacing at most three. Deleting it preserves balance, and every surviving periodic balanced set is again a rational Beatty set. Induction determines the surviving binary scales, while a two-sequence disjointness criterion forces the deleted density to be the next binary scale. This yields the asserted density pattern.An eventual Beatty representation of balanced binary indicators extends the classification to one-sided balanced sequences on at least three letters with distinct positive densities, proving the balanced-sequence conjecture of Altman, Gaujal, and Hordijk.

Let $E/\mathbb{Q}$ be a rational elliptic curve whose torsion group grows over a quadratic field $K$. In a previous paper, a relation between the primes of the conductor of $E$ and the primes that divide the discriminant of $K$ was shown. In the present paper, we go further in this study and we compute the ramification of the field $K$, when a torsion point of prime order $\ell$ is added, in terms of invariants of the curve. Concretely, the conductor, the prime $\ell$ and the Kodaira symbol at the primes of bad reduction.

We construct additive rank-metric and sum-rank-metric codes from Drinfeld modules by restricting bounded-degree morphisms to prime-to-characteristic torsion. For supersingular Drinfeld modules of rank $r$ in characteristic $\mathfrak{p}$ of degree $d$, the stabilization formula for morphism spaces yields rank-metric codes of $\mathbb{F}_q$-dimension $mrt-c$ and minimum distance $r-t+1$, where $c=r(r-1)(d-1)/2$. Simultaneous restriction to $\ell$ distinct degree-$m$ torsion modules gives additive sum-rank codes of the same dimension and minimum distance at least $\ell r-t+1$. Their normalized Singleton defects tend to zero, while in characteristic $(T)$ the module $\phi_T=\tau^r$ makes the defect vanish and produces an explicit MSRD family. We identify this family with a skew Chinese remainder theorem code supported on central skew polynomials and prove that its poly-skew weight is exactly $m$ times its sum-rank weight. This gives a specialized Singleton-type bound and a polynomial-time unique decoder up to the full sum-rank unique-decoding radius. We also derive a Welch-Berlekamp-type filter equation for the general supersingular sum-rank construction; it becomes an effective decoder whenever bases of the relevant morphism spaces and the restriction maps are computable.

In this paper, we construct confluence relations for $q$-analogues of multiple zeta values by adapting the classical construction to the $q$-setting. Our construction uses $q$-analogues of multiple polylogarithms depending on an auxiliary variable $z$, functional relations arising from their $q$-differential equations, and the (regularized) limit as $z$ tends to $1$. We prove that the resulting family of relations contains the stuffle product relations and, assuming a certain conjectural identity, the duality relations for both the Bradley–Zhao and Schlesinger–Zudilin models.

Motivated by mirror symmetry, we study exponential sums over the $\mathbb{F}_q$-points of the $A_1$ Coulomb branch. We show that their fiberwise structure is governed by polynomial Gauss sums, providing a geometric realization of Selberg character sums studied by Evans. Using Evans's evaluation, we obtain explicit formulas for the Coulomb-branch exponential sums as products of classical Gauss sums.

We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.

We establish a regularized spectral decomposition of the squared magnitude of a maximal degenerate Eisenstein series as a Schwartz distribution on $\mathrm{GL}_n$. Although the original problem is on $\mathrm{GL}_n$, its spectral components are described entirely by the automorphic spectrum of $\mathrm{GL}_2$. As an application, we prove a partial reciprocity formula relating the second moment of $\mathrm{GL}_n\times\mathrm{GL}_n$ Rankin–Selberg central $L$-values and a mixed moment of $L$-values of $\mathrm{GL}_2$. We also prove, assuming the generalized Ramanujan conjecture, an asymptotic formula for the second moment of the above Rankin–Selberg $L$-functions over a conductor-aspect family with an error term of square-root-cancellation strength.

We construct permutation polynomials of the form x + \gamma Tr^{q^2}_q(h(x)) over the finite field F_{q^2}. More precisely, we present two families ofpermutation polynomials whose coefficients range over all elements of the field rather than being restricted to a proper subfield. We also compute the compositional inverses of both families. Our techniques involve the evaluation of certain Kloosterman sums together with the analysis of some equations over finite fields, and we believe these can be of independent interest.

The study of the largest prime factor in polynomial sequences can be traced back at least to the late 19th century in the work of Störmer. Mahler (1933) and Chowla (1934) proved that the largest prime factor of $n^2+1$ grows at least as fast as $\log_2 n$. In 2023 we improved this to $(\log_2 n)^2/\log_3 n$. In this note we show the lower bound $(\log_2 n)^2/\log_4 n$, and that when this bound is nearly sharp it also holds for many prime factors of $n^2+1$.

In this paper we study the motivic Chabauty–Kim method, which aims to determine the set of $S$-integral points of $\mathbb{P}^1\smallsetminus \{0,1,\infty\}$, over cyclotomic fields. We focus on the case $K=\mathbb{Q}(\zeta_8)$ and $S=\left\{(1-\zeta_8)\right\}$, where we obtain explicit polylogarithmic Kim functions up to depth $4$ and verify Kim's Conjecture for several primes. We also observe and explain that the Chabauty–Kim locus for the polylogarithmic quotient contains, in addition to the $S$-integral points, certain exceptional points arising from roots of unity in $\mathbb{Q}_p$.

Let $K_1,\cdots,K_r$ be fixed number fields, and let $L$ be the compositum of their Galois closures. Assuming GRH for $\zeta_L$, we prove a sharp lower bound for products of shifted Dedekind zeta functions on the critical line, for arbitrary fixed positive real exponents and uniformly for shifts of size at most $T/2$. The correlation factor is expressed as a product of Dedekind zeta functions of the fixed fields of double-coset stabilisers in $\textrm{Gal}(L/\mathbb{Q})$. Combined with the corresponding upper bound by the authors, determines the order of magnitude of these shifted moments for both Galois and non-Galois fields.

Let $E$ be an $n$-dimensional rigid adelic space over a number field $K$. We study the minimum number $g_E(R)$ of proper $K$-subspaces needed to cover the projective height ball of radius $R$, together with the maximum cardinality $h_E(R)$ of a subset in linear general position. We show that, once $R$ is sufficiently large compared with the last Roy–Thunder minimum of $E$, both quantities have order $\Psi_E(R)^{[K:\mathbb Q]}$, where $\Psi_E(R)$ is an explicit expression in the Roy–Thunder minima. The comparison constants are effectively computable and uniform in $E$. For the standard adelic space $K^n$, this gives order $R^{[K:\mathbb Q]n/(n-1)}$.

We work over a field of characteristic zero - primarily over an algebraic closure of the field of rational numbers. We prove a necessary and sufficient condition for a vector field whose coefficients are rational functions separating variables to be algebraically integrable, that is, the subring of rational functions killed by this vector field, its ring of first integrals, is of maximal possible dimension. We do it arithmetically by reducing the vector field modulo almost all primes. In particular, we verify the generalized Grothendieck–Katz p-curvature conjecture for foliations defined by these vector fields. Finally, we use the outcome of that verification to provide explicit formulas for coefficients of all algebraically integrable vector fields separating variables, and their first integrals.

Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels. We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position $p^k/X$, where $X$ is the conductor scale. For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed $k$ under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and Müller. Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.

We improve the best known result on the number of variables needed for the smooth Hasse principle for homogeneous forms of degree \(d\geq5\).

The quantum Rabi model occupies a distinguished role in the study of quantum light and matter, describing the most fundamental interactions. Elucidation of its spectral properties, along with those of its generalizations, is necessary for quantum optics and applications in areas such as quantum information technologies. In this paper, we explore a precise description of energy degeneracy of the two-photon asymmetric quantum Rabi model and the nature of the system's symmetries; as a result, we demonstrate that these have significant relationships with the structure of certain algebraic curves (hyperelliptic curves) and number theory. Concretely, we first prove the existence of degenerate eigenvalues using the monodromy data of the Fuchsian ODE, which draws the eigenvalue problem of the system, and the existence of associated symmetry operators, previously discovered and studied heuristically in physics. Subsequently, we give a complete characterization of spectral degeneracies and present a series of conjectures showing that the degeneracy may determine the entire energy spectral structure. More precisely, for sufficiently strong interaction, we obtain an excellent approximation of the energy curves by the zero locus of a two-variable polynomial associated to degenerate eigenstates. In addition, we postulate an explicit description of the relation between the missing symmetry at the Hamiltonian level (hidden symmetry) and the degeneracy of the two-photon quantum Rabi model.

In this paper, we establish a summation formula associated with the generalized Apostol-Bernoulli functions. This formula unifies the Euler-Maclaurin summation formula, the Euler-Boole summation formula, and the character analogues of these two formulas. We also apply it to obtain the general power sum formula and the special values of Berndt's generalized $L$-function at non-positive integers.

Let \( \mathcal{S}_{g,n} \subset \mathcal{M}_g \) be the locus of curves of genus \( g \geq 2 \) admitting a model \( y^n = h(x) \) with \( h \) separable; such curves $C$ have a cyclic group \( C_n \leq \operatorname{Aut}(C) \) of order \( n \) with \( C/C_n \cong \mathbb{P}^1 \). % We give an algorithm which, given an absolutely irreducible plane model \( F(x,y) = 0 \) of a curve \( C \) over a field \( k_0 \) of characteristic zero, decides for which \( n \) the curve lies in \( \mathcal{S}_{g,n} \) and returns a model \( y^n = h(x) \) together with the birational transformation to it.

We introduce the notion of Hodge–Tate splitting for schemes in positive characteristic. For a smooth variety $X$ over a perfect field $k$ of characteristic $p > 0$, we say that $X$ is Hodge–Tate split if the natural morphism $\mathcal{O}_X \to F_*\Omega^\bullet_{X/k}$ induced by the absolute Frobenius admits a splitting in $\mathcal{D}_{\mathrm{qcoh}}(X)$. We prove that this condition is equivalent to the existence of a decomposition $F_*\Omega^\bullet_{X/k} \simeq \bigoplus_{i=0}^{\dim X}\Omega^i_{X/k}[-i]$ of its de Rham complex. Consequently, for smooth projective varieties, Hodge–Tate splitting implies Akizuki–Nakano vanishing and the $E_1$-degeneration of the Hodge-to-de Rham spectral sequence. Furthermore, we establish criteria and permanence properties for Hodge–Tate splitting and use them to construct many new examples of varieties whose de Rham complexes decompose. These include blow-ups of quasi-$F$-split varieties along strata of simple normal crossings divisors, complete intersections in toric varieties, and linearly reductive quotients of Hodge–Tate split varieties. Among these examples, we obtain smooth projective varieties whose Hodge-to-de Rham spectral sequences degenerate at $E_1$, whereas their Hochschild–Kostant–Rosenberg spectral sequences do not degenerate. As an application in mixed characteristic, we prove an Akizuki–Nakano-type vanishing theorem for smooth projective globally $+$-regular varieties over the Witt ring of a perfect field.

Kaneko's val function is defined as the normalized cycle integral of the elliptic modular $j$-function along closed geodesics on the modular surface. We prove that, when primitive hyperbolic conjugacy classes are ordered by geodesic length, its values concentrate at the single point 720. More generally, an analogous concentration result holds for every weakly holomorphic modular function $f$ of weight 0, with the concentration point given by Atkin's inner product $(f, 1)_{\mathrm{At}}$. The proof combines an equidistribution theorem following Pollicott with the ergodicity of a continued-fraction suspension flow and uses the decomposition formula of Bengoechea-Imamoḡlu to construct a bounded continuous observable.

We show an explicit version of the Van der Corput truncated Poisson summation formula (B-process). By using refined explicit exponential sum estimates, this improves the error term of previous explicit results by Patel and Yang (2024) and Arias de Reyna (2024). As an application, we obtain fine explicit estimates for the error terms in approximate functional equations for the Riemann zeta function, improving on previous explicit results of Simonič (2020) in certain ranges.

We study the rational points on the star curve $X_0^*(N) := X_0(N)/W(N)$, the quotient of the classical modular curve $X_0(N)$ by the full group of Atkin–Lehner involutions, for squarefree levels $N$. Rational points on $X_0^*(N)$ parameterize $\mathbb{Q}$-curves, i.e.\ elliptic curves $E/\overline{\mathbb{Q}}$ that are isogenous to all of their Galois conjugates. Elkies conjectures that $X_0^*(N)$ has only CM or cuspidal rational points for all large enough $N$. We call any other rational points "exceptional". In this article, we provide new examples of exceptional points in genus 3 and 4, and we give evidence that no exceptional points exist in genus $g \geq 5$. Moreover, we investigate the underlying geometric reasons that might "explain" why these exceptional points arise in the first place, in the vein of Ogg and Mazur. In particular, we propose geometric explanations for Galbraith's exceptional points on $X_0^*(137)$ and $X_0^*(311)$.

In this paper we study the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of a direct product of symmetric groups. We first treat, in detail, the case of a product of two symmetric groups, and establish a decomposition formula for the resultant of such systems. We then show that this decomposition, together with the underlying combinatorics, extends to an arbitrary (finite) direct product of $r$ symmetric groups. Thanks to these decomposition formulas, we prove that the discriminant of a multivariate homogeneous polynomial invariant under a direct product of $r$ symmetric groups splits into a product of resultants of smaller size that are easier to compute.

For a $p$-adic field of any residual characteristic, we determine irreducible decompositions of the Weil representation restricted to the maximal open compact subgroups and the Iwahori subgroup by using the Schrödinger model. In particular, these decompositions show that the restrictions are multiplicity free.

For positive integers $n$ and $m$, we study the actions of the groups of tame automorphisms of the $n$-dimensional affine spaces over finite fields on ordered subsets of $m$ points. Our primary interest lies in constructing tame automorphisms that take one ordered sequence to another and in proving upper and lower bounds on the maximal complexity of such automorphisms. Our preferred measure of complexity of an automorphism is the maximum of the degrees of the polynomials that define it and its inverse. Using methods and results from various branches of mathematics, including the theory of symmetric groups, affine geometry over finite fields, polynomial interpolation, combinatorics of projective spaces over fields, and polynomial automorphisms, we obtain a wide variety of qualitative and quantitative results.