Papers

We prove that each of the inequalities $P^{+}(p+1)>P^{+}(p-1)$ and $P^{+}(p-1)>P^{+}(p+1)$ holds for a positive proportion of primes. The argument combines a logarithmic balance identity with a uniform Selberg sieve over a thin cofactor strip. The same transfer mechanism applies to either shift and converts a one-sided large-prime-factor reservoir into an unconditional ordering result in both directions.

Let $q_n$ denote the $n$th product of two distinct primes. By completing a square in Sono's sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every $\varepsilon>0$ and all sufficiently large integers $\rho$, \[ \liminf_{n\to\infty}(q_{n+\rho}-q_n) \leq \exp\left((\pi+\varepsilon)\sqrt\rho\right). \]

Let $G$ be a reductive group over a number field $F$. We define $C^*_{\rm aut}(G(\mathbb A))$, the automorphic $C^*$-algebra of $G$, to be the $C^*$-algebraic image of the full automorphic representation of $G(\mathbb A)$ on $L^2(G(F)\backslash G(\mathbb A))$. Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective $*$-homomorphism \[ C^*_{\rm aut}(G(\mathbb A)) \longrightarrow \bigoplus_{[M,\sigma]} K_{C_0(\widehat{A_M})} \left( \operatorname{Ind}_P^G C_0(\widehat{A_M},H_{M,\sigma}) \right)^{W(G,M,\sigma)}, \] where $[M,\sigma]$ ranges over the associate classes of discrete Levi data. When $G$ is $\operatorname{GL}(n)$ or an inner form of it, we prove that this map is an isomorphism of $C^*$-algebras.

We solve Conjecture 1.6 of Furio and Lombardo [On $7$-adic Galois representations for elliptic curves over $\Q$, Proc.\ London Math.\ Soc.\ (3) 133 (2026), e70193.]. As a consequence, we complete the classification of $7$-adic Galois images for non-CM elliptic curves over $\Q$. This is the first solved case in Problem 6.2 from Balakrishnan [Chabauty and beyond: Explicit p-adic methods for rational points, Proceedings of the International Congress of Mathematicians 2026 - Volume 3: Invited Lectures (Sections 1-4), 321-341, 2026]. The proof is the result of a long AI-training period by the author from June 2026 to September 2026. The author has been working on this project alone without the help of any other.

Let $\Gz$ be a cofinite Fuchsian group whose quotient $\Gz\backslash\HH$ has genus one and a single cusp, normalized to be at $\infty$ with width one. Let $\Xg$ be its smooth compactification, which is an elliptic curve over $\CC$. Let $x,y$ be canonical generators of the function field on $\Xg$, which have poles of order $2,3$ respectively at the cusp and which {satisfy} a generalized Weierstrass relation $y^2+(a_1x+a_3)y=x^3+a_2x^2+a_4x+a_6$. Let $f$ be the unique (up to scale) weight two cusp form for $\Gz$, normalized so that the associated holomorphic differential is $\omega=dx/(2y+a_1x+a_3)$. With all this, we prove that the generating function identity \[ \frac{\bigl(y(\tau)+y(z)+a_1x(z)+a_3\bigr)f(z)}{x(z)-x(\tau)}=\sum_{m\ge0}\Phi_m(\tau)q_z^m \] defines a family $\{\Phi_m\}_{m\ge0}$ of weakly holomorphic modular functions on $\Xg$ such that the set $\{\Phi_0\}\cup\{\Phi_m\}_{m\ge2}$ is a canonical basis of the space $M_{0,\Gz}^{!,\infty}$ of weakly holomorphic modular functions on $\Xg$ with poles supported only at the cusp. As an application, by comparing the generating function of the family $\{\Phi_m\}_{m\ge0}$ with the generating function of the Niebur–Poincaré series, we show that the generating function of the special values of the Kloosterman zeta function at $1$ is the holomorphic part of a certain weight two harmonic Maass form up to an additive constant.

Let $\mathcal G$ be a split semiabelian variety of dimension $n \ge 2$ and let $\mathcal C \subset \mathcal G$ an irreducible curve, where everything is defined over $\overline{\mathbb{Q}}$. We prove that, under the natural hypothesis that $\mathcal C$ is not contained in any proper algebraic subgroup of $\mathcal G$, the set of points $P$ of $\mathcal C$ lying in an algebraic subgroup $H$ which intersects the curve tangentially at $P$ is finite. This generalizes a result for curves in 2-dimensional tori by Marché and Maurin. We apply these results to square-freeness of certain geometric divisibility sequences.

For an integer solution of $y^2=x^3+k$ with $x,y\ge 1$ and $k\ne 0$, the integers $x^3$, $|k|$ and $y^2$ form a triple in which one term is the sum of the other two. When $\gcd(x,y)=1$ this triple is coprime, and the $abc$ conjecture predicts that its quality is asymptotically at most $1$. Following Müller, Taktikos and de Weger, we factor this quality through the product $P=xy|k|$ into the approximation gain $G_a=\log\max(x^3,y^2)/\log P$ and the power gain $G_p=\log P/\log\rad(xy|k|)$, which is at least $1$. We show that the weak form of Hall's conjecture, which asks that $|k|>x^{1/2-\delta}$ for every $\delta>0$ outside finitely many solutions, is equivalent to the asymptotic bound $G_a\le 1$. More generally, for $6/11\le\kappa<6/5$, an asymptotic bound $G_a\le\kappa$ gives $|k|>x^{3/\kappa-5/2-\delta}$ for every $\delta>0$ outside finitely many solutions, and for $3/4\le\kappa<6/5$ the two statements are equivalent. Since the approximation gain never exceeds the quality, an $abc$ inequality with exponent $1\le\kappa<6/5$ gives the same bound for primitive solutions. The case $\kappa=1$ extends to all solutions and recovers the classical implication from the $abc$ conjecture to weak Hall. We also relate small values of $|k|$ to large partial quotients of $\sqrt x$.

Let $f$ be a Steinhaus random multiplicative function and let $g$ be a polynomial of degree $d$. Write $S_N=N^{-1/2}\sum_{n\le N}f(n)\,e(g(n))$. We prove a quantitative dichotomy for the moments of $S_N$: for each integer $s\ge 2$, either $\mathbb{E}|S_N|^{2s}$ is close to the Gaussian moment $s!$, or the coefficients of $g$ can be approximated by rationals with a small denominator. In particular, if the coefficients of $g$ satisfy a Diophantine condition, then $S_N$ converges in law to a complex normal distribution with mean $0$ and variance $1$. Lean 4 code for the proofs is provided, for convenience of verification.

Let $d\ge2$ and let $\mathbf w=(w_1,\ldots,w_d)$ satisfy $w_1\ge\cdots\ge w_d>0$ and $\sum_i w_i=1$. Set $s_*=d-(1+w_1)^{-1}$. We prove that there exist constants $C_{d,\mathbf w}>0$ and $\varepsilon_0=\varepsilon_0(d,\mathbf w)>0$ such that for all $0<\varepsilon<\varepsilon_0$, $$ \dim_H\operatorname{DI}_{\mathbf w}(\varepsilon)\le s_*+C_{d,\mathbf w}\sqrt\varepsilon. $$ Together with the lower bound of Kim–Park, this gives the exact formula $$ \dim_H\operatorname{Sing}(\mathbf w)=s_*. $$

In this paper, we study a numeration system with a non–integer base $a>1$ over the binary alphabet $A=\{0,1\}$: $$\left[0;\frac{r}{a-1}\right]\ni x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n} \equiv \Delta^{r_a}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}, \quad \alpha_n\in A.$$ We investigate the geometry of $r_a$–representation of numbers, including the geometric interpretation of digits, the structure of cylinder overlaps, and the associated metric properties. The main object of our study is a nowhere monotone function of unbounded variation defined by $$f\left(x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{2^n}\right) =\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n},$$ where the classical binary representation of the argument is assumed not to end with the infinite period $(1)$. We derive a system of two functional equations satisfied by $f$. We prove that the function has fractal level sets and a self–affine graph. We also evaluate the integral of $f$ over the interval $[0,1]$.

We give a new proof that if the natural numbers are colored in two colors, then there exists a monochromatic Pythagorean triple. The proof was found by ChatGPT Astra.

In this article, we study derived Hecke algebras with coefficients in torsion-free profinite rings regarded as discrete rings. We compare each such algebra with the degreewise inverse limit of the derived Hecke algebras with coefficients in finite rings and with its subalgebra consisting of elements with uniformly finite double-coset support. Under cohomological comparison hypotheses, the natural map to this subalgebra has image given by its degree-zero part and its positive-degree torsion. We describe the kernel in terms of continuous cohomology and prove that it is a divisible square-zero ideal annihilated by every positive-degree element. These comparisons are compatible with Yoneda products, and the action on torsion Ext classes factors through the image. We compute the resulting algebras for the additive group $\mathbb{Q}_p$ and determine the underlying graded abelian groups for $(\operatorname{GL}_2(\mathbb{Q}_p),\operatorname{GL}_2(\mathbb{Z}_p))$ with coefficients in $\mathbb{Z}_p$, for $p>3$. The examples exhibit distinct effects of the coefficient topology and the support condition.

We construct some part of the plectic Galois action on the locally analytic $p$-adically completed cohomology of Hilbert modular varieties, predicted by the plectic conjecture of Nekovář-Scholl. As a byproduct, we prove a vanishing result for the completed cohomology of Hilbert modular varieties below middle degree. To do this, we make use of the partial Sen operators constructed earlier by a geometric method.

We prove that the pushforwards of Gauss–Manin connections on smooth rigid analytic varieties along Zariski-open immersions are coadmissible and holonomic over Ardakov–Wadsley's sheaf D-cap of infinite order differential operators. This may be viewed as a rigid analytic variant of the classical theorem of Griffiths, Deligne, and Katz that Gauss-Manin systems have regular singularities after compactification. The proof combines p-adic de Rham comparison and Diao–Lan–Liu–Zhu's logarithmic Riemann–Hilbert correspondence with extendability results for log-connections with nilpotent residues. In the algebraic snc case, we also establish weak holonomicity via a p-adic Bernstein-Sato criterion of Bitoun and the first author.

Let $\pi$ be an irreducible generic representation of $\operatorname{GL}_n(\mathbb{C})$. We classify the exceptional poles of the associated Asai $L$-function $L(s,\pi,\mathrm{As})$ and give a representation-theoretic characterization of them. When $\pi$ is in general position, we further express $L(s,\pi,\mathrm{As})$ in terms of the exceptional $L$-factors attached to the irreducible constituents occurring in the derivatives of $\pi$.

By a result of Bogomolov and Zarhin, a K3 surface $X$ over a number field $L$ has ordinary reduction at a density 1 set of primes after passing to a finite extension of $L$. In this paper, we refine this result for non-CM K3 surfaces whose transcendental Hodge structure has endomorphism field $F$ abelian over $\mathbb{Q}$. We further assume a condition on the connected components of the $l$-adic monodromy group of $X$, and, when $F$ is totally real, a parity condition on the rank of the transcendental lattice over $F$. Under these assumptions, we prove that the set of primes of $L$ at which $X$ has $\mu$-ordinary reduction has density $1$. As a corollary, the set of primes of $L$ at which $X$ has ordinary reduction has density $1/[FL:L]$. We include explicit examples of K3 surfaces satisfying these conditions.

For a fixed modulus h and residue class c, we study the logarithmic mass of primes p in a fixed power band for which the exponent of p in n! is congruent to c modulo h. We establish the asymptotic ((b-a)/h + o(1)) log(n!) for primes satisfying n^a < p <= n^b, where 0 <= a < b <= 1. The argument combines a fixed-order derivative estimate, Vaughan's identity, and finite Fourier approximation of a discontinuous valuation function. We also derive a necessary capacity condition for infinite families of solutions to polynomial-factorial equations P(x) = n!.

Bringmann, Han, Heim, and Kane recently investigated periodic sign patterns in the Fourier coefficients of weakly holomorphic modular forms arising from eta-quotients and posed numerous conjectures in the appendix to their work. Using classical theta-function identities, dissections, and coefficientwise positivity, we prove 72 of these conjectural sign patterns. The eta-quotients considered here have periods 2, 3, 4, 5, 6, 8, and 12. Our proofs are elementary at the level of $q$-series and yield explicit generating functions for the coefficients in each relevant residue class.

We develop a hypergeometric framework for elliptic integral moments organized around Mishev's completed Saalschützian $L$-function and its $W(D_5)$ symmetry. The Bailey–Mishev very-well-poised ${}_7F_6(1)$ representation places complementary elliptic integral moments and harmonic hypergeometric sums in a common parameter space. We first give collision-based hypergeometric derivations of the three complementary-modulus moment families involving $K'^2$, $E'K'$, and $E'^2$. The $E'^2$ identity is also recovered independently from a Barnes-contiguous relation mirroring the differential system of the elliptic integrals. We then derive a hypergeometric evaluation of the harmonic-series representation of the four-Bessel moment $s_{4,0}$. At symmetric parameter points, Coxeter symmetry forces selected Taylor coefficients to vanish along symmetry-adapted paths, producing families of harmonic and special-value identities involving values of the Riemann zeta function and the same Bessel period. The resulting picture shows that these elliptic-moment, harmonic-sum, and special-value identities arise from a common $W(D_5)$ symmetry of generalized hypergeometric functions.

We study piecewise polynomial functions $\delta_k(c)$ that appear in the conjectured asymptotics for the variance of short-interval sums of von Mangoldt convolutions $\Lambda^{*k}$. By a theorem of Kuperberg and Lalín, a unitary matrix integral involving a sum of trace products governs the large-$q$ limit of the function-field analogue of this variance. We identify $\delta_k(c)$ as the leading-order coefficient of this integral when the total trace degree and matrix size grow with a fixed ratio $c$. Through an asymptotic analysis of two lattice-point representations for the matrix integral, we obtain explicit forms of $\delta_k(c)$. A finite spline expansion and a contour-integral representation expressing $\delta_k(c)$ as the inverse Laplace transform of a two-weight block Hankel determinant are also obtained.

We demonstrate a modification of the high-low method of Guth-Maldague-Wang that yields a decoupling theorem which gives the sharp estimate for the discrete restriction constant for the parabola and also a diameter-free bound for the three-fold additive energy of rational points on the parabola and circle. These results were recently obtained by Deng-Fan-Guo-Guo-Luo and Cushman-Demeter-Wu, respectively. The novel features in this argument are an orthogonal expansion of $F^3$ and a new square function that also keeps the wavepackets discarded after pruning, resulting in new and more efficient high and low lemmas.

For cocompact Fuchsian groups $\Gamma \leq \hbox{PSL}(2, \mathbb{R})$ we improve Selberg's bound for the error term of the counting function in the hyperbolic lattice counting problem, achieving an essentially optimal upper bound for the error term \begin{equation*} E(X;z, w) = O(X^{1/2+\varepsilon}) \end{equation*} for almost every pair of points $z,w$. We extend this pointwise result for cocompact lattices acting on the $n$-dimensional real hyperbolic space $\mathbb{H}^n$. Moreover, in the $2$-dimensional case we study the second moment of the error term of the counting function. We improve the upper bounds of Chamizo and Cherubini for almost all pairs $z,w$, but we disprove a conjecture of Phillips and Rudnick for the case of cocompact Fuchsian groups.

We obtain improved omega bounds for the error term in the Piltz divisor problem over number fields. Our proof combines Lamzouri's resonance argument with a counting theorem for nonnegative multiplicative functions. The counting theorem gives the estimates needed for the divisor coefficients over a number field and allows us to remove the factor involving the third iterated logarithm from the earlier bounds.

We prove that any proper Shimura variety admitting a limpid integral model in the sense of Madapusi Pera becomes a perfectoid space at infinite level at $p$. Our method provides a new geometric approach that circumvents the need for a universal abelian scheme, making it applicable beyond Shimura varieties of Hodge type. Inspired by Scholze's proof of the perfectoidness of the Siegel modular variety, we construct analogues of the canonical subgroup and the anticanonical tower within a strict neighborhood of the ordinary locus. To achieve this we utilize the geometry of the universal $G$-aperture over the limpid integral model. Because every Shimura variety admits a limpid integral model for sufficiently large primes, our result implies that all proper Shimura varieties are globally perfectoid at infinite level for big enough primes.

Colossally abundant numbers have large sums of divisors relative to their size. They also admit a geometric description using supporting lines of a planar convex hull. Starting from this description, we change coordinates to obtain nested subsets of these numbers, which we call colossally abundant numbers of higher order. Their definition does not depend on the Riemann hypothesis. We give conditions under which Robin's inequality holds on any one of these subsets if and only if the Riemann hypothesis is true. Some of the resulting families have infinite sets at every level but an empty intersection. For two classes of coordinates, we determine how fast the least members grow. We also prove that every fixed positive integer divides all members of sufficiently high order. Under a concavity assumption, the least members form a divisibility chain; for power abscissas, successive quotients have unbounded numbers of prime factors, with an explicit limsup growth rate. In a second class of families, every global maximum of the normalized divisor sum is retained, and the least members tend to infinity if and only if the Riemann hypothesis is true. We give numerical examples and explain how the Ramanujan–Nicolas bounds produce terminating chains.

Fix a prime $\ell$. Let $K = \mathbb{F}_q(t)$ be a global function field such that $\gcd(q,6) = 1$ and $q \equiv 1 \pmod \ell$. Let $E$ be a non-isotrivial elliptic curve over $K$. Given a fixed monic polynomial $Q$ over $\mathbb{F}_q$, and assuming some mild conditions on $E$, we show that the rank of $E$ does not change with respect to a positive proportion of $\mathbb{Z}/\ell \mathbb{Z}$ extensions $K(\sqrt[\ell]{f})/K$, as $f$ varies over the set of monic polynomials over $\mathbb{F}_q$ such that $f \equiv A \pmod Q$ for any given polynomial $A$. We obtain this result by combining analytic and probabilistic techniques to study the distribution of certain prime Selmer groups of auxiliary abelian varieties constructed from these polynomials $f$.

We investigate arithmetic and combinatorial properties of the function $a(n)$, which is the signed number of partitions of $n$ into exactly two distinct part sizes, each occurring an odd number of times. We prove that $a(n)\ge 0$ for all $n$, establishing a positivity phenomenon for a signed partition function arising from a double Lambert series. Furthermore, we show that $a(n)$ satisfies nontrivial divisibility properties modulo $3$. The results reveal unexpected arithmetic regularity in a family of partition functions defined by odd multiplicity constraints and restricted support.

Let $\mathbb{Z}_K$ denote the ring of integers of the number field $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $\mathbb{Z}_K=\mathbb{Z}[\theta]$. A polynomial $f(x)\in\mathbb{Z}[x]$ is called reciprocal if $f(x)=x^{\deg(f)}f(1/x)$. In this article, we establish necessary and sufficient conditions for the monogeneity of reciprocal polynomials. As an application, we obtain an alternative and much simpler proof that the maximal real subfields of cyclotomic fields are monogenic.

We prove an inverse theorem for the Gowers uniformity norm $\mathsf{U}^k(\mathbb{F}_p^n)$ with quasipolynomial bounds in the case when $p \geq k$. The inverse theorem follows from a quasipolynomial structure theorem for Freiman multihomomorphisms, which are a natural generalization of Freiman homomorphisms to maps of several variables. The proof of the structure theorem for Freiman multihomomorphisms is the central result of the paper and rests on three main ingredients: algebraic regularity method, abstract Balog-Szemerédi-Gowers theorem and the theory of multilinear maps defined on multilinear varieties. The last ingredient originates from an earlier work of Gowers and the author, and is significantly expanded in this paper. In particular, once the theory of such maps is in place, the proof of the structure theorem for Freiman multihomomorphisms is relatively short, especially compared to the previous quantitative results in the inverse theory of Gowers norms.

For $d\ge2$ the $q$-deformed modular group specialized at a primitive $d$-th root of unity is the triangle group of type $(2,3,d)$. Using this we determine the fractions $r/s$ for which the $d$-th cyclotomic polynomial divides the denominator $S_{r/s}(q)$ of the $q$-deformed rational number $[r/s]_q$. They form the orbit of $\infty$ under the normal closure of the translation $z\mapsto z+d$ in $\mathrm{PSL}(2,\mathbb{Z})$. This proves a conjecture of Byakuno, Ren and Yanagawa, and shows that the congruences $s\equiv0$ and $r\equiv\pm1$ modulo $d$ characterize the divisibility exactly for $d\le5$. For $a\in\{2,3,4,6\}$ and $n>5a^2$ prime to $a$ we show that $S_{a/n}(q)$ is irreducible up to cyclotomic factors. Together with a computer check for small primes, this confirms a conjecture of Kogiso, Ren, Wakui, Yanagawa and the author for every prime $p$ and every $r$ prime to $p$ with $r\equiv\pm a$ or $ar\equiv\pm1\pmod p$.