Papers

We give an explicit local description of ray class refinements in cyclotomic Iwasawa theory. For distinct primes $p\geq5$ and $\ell$, and even $2\leq k\leq p-3$, we prove that the $\omega^{1-k}$-component of the $p$-primary ray class group of $\mathbb{Q}(\zeta_p)$, with squarefree modulus supported above $\ell$, is nonzero if and only if $p\mid\ell(1-\ell^k)B_k$. In the cyclotomic tower, we determine the local modules and their correction modulo the diagonal global Tate module, obtaining an explicit determinant formula. Their Frobenius Euler factors, combined with the Mazur–Wiles theorem, yield a ray class form of the Iwasawa main conjecture for nonexceptional odd characters. We also obtain exact class number valuation comparisons at every layer, including those before stabilization, recover Finat's formula for $\ell=2$, and deduce refined $\mu$-vanishing from Ferrero–Washington. The cyclotomic character explains the Wieferich exception, while existing Eisenstein congruence theorems connect the divisibility criteria with prime-level Ramanujan-style congruences.

Let ${\mathbb P}^1_{\mathfrak q}$ be the weighted projective line with coprime weights ${\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. The weighted Veronese map $\Phi\colon {\mathbb P}^1_{\mathfrak q}\to {\mathbb P}^1$ satisfies $h\circ\Phi=L\,{\mathfrak s}$. Pulling back Roth's theorem gives the optimal exponent $2L$ for the full rational orbit set. For points represented by coprime integral coordinates, counting suggests the smaller exponent $q_0+q_1$. We formulate this as a conjecture and prove an $S$-adic radical form under the $abc$ conjecture. For ${\mathfrak q}=(2,3)$ the resulting inequality at the target $[1:1]$ is the coprime Hall-type inequality, and Danilov–Elkies examples show that the exponent $5$ is sharp. We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. A weighted $abc$ formulation is introduced; in signature $(3,2)$ finitely many coefficient pairs are equivalent to $abc$, and weighted Belyi maps transfer this input to arbitrary algebraic targets. Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction.

We study hyperelliptic curves on $K3$ surfaces. We show that any number of general points on a $K3$ surface of Picard rank one can be connected by a hyperelliptic curve. As a by-product we recover the Beauville–Voisin decomposition of the small diagonal, together with some new bounds on its torsion order in the integral Chow group. On the arithmetic side, given a $K3$ surface $X$ of geometric Picard rank one defined over a number field $k$, we show that there exists a finite extension $k\subset k'$ such that quadratic points are dense in $X_{k'}$, extending a theorem of Corvaja–Zucconi to $K3$ surfaces of arbitrary degree.

Recently, Matomäki, Pandey, Pliego, Teräväinen and Wang improved the Baker–Harman–Pintz estimate of trilinear sums over short intervals of length $x^{\theta}$, which naturally reduces to estimating mean values of products of three Dirichlet polynomials. Based on recent works of Guth and Maynard and Chen, Gupta and Li on new large value estimates for Dirichlet polynomials, they improved the Baker–Harman–Pintz estimate for $\theta > \frac{3}{5}$. In this paper, we use their method to improve the Baker–Harman–Pintz estimate for $\frac{1}{2} < \theta < \frac{4}{7}$ and get new results for all $\frac{1}{2} < \theta < 1$. Our result yields improvements on many problems, including the work of Li on primes in short intervals of length $x^{0.52}$.

We study digit-appending walks in base $b\geq 2$ along integer linear recurrences with minimal characteristic polynomials having a unique simple dominant root $\alpha$ that lies outside the unit circle. We prove that if $\alpha$ and $b$ are multiplicatively independent, then no infinite walk can have uniformly bounded appended digit lengths. In contrast, we show that when the appended digit length can be unbounded, the same independence hypothesis implies the existence of infinite walks.

We study the nonlinear phenomenon of the asymptotics of higher-order derivatives of the Gram function $t_u$. We also prove the uniform distribution and continuous uniform distribution modulo one of $t^n_u$ for $n = 1, 2, \dots$. We use AI to formalize our proofs in Lean.

We show how the methods of delta geometry can be used to bound the degree of points in the Zilber–Pink locus of a subvariety of an abelian variety, if the varieties are defined over a number field and we restrict to points with bounded ramification at a fixed prime.

Let $G$ be a split connected semi-simple group over a local non-archimedean field $F$. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack $\bpo$ of $G$-bundles on ${\mathbb P}^1$ endowed with a trivialization at $0$ and $\infty$. This space is a representation of $G(F)\times G(F)$ which contains the regular representation of $G(F)$. We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation $\pi$ of $G(F)$ defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of $G(F)$ with an Iwahori fixed vector defines a ${\mathbb C}$-point on the Calogero-Moser space associated to $G$. We also show that for $G=PGL(N)$ the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of $\pi$ on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when $F$ is a finite field.

The Frobenius problem asks, given two positive, relatively prime integers $a$ and $b$, what is the maximum integer that cannot be expressed as a linear combination of $a,b$ using non-negative coefficients? This problem is resolved. However, if we add the restriction that the coefficients themselves must also be relatively prime, we create an open problem with connections to the Stern and Farey sequences and the Jacobsthal function. We prove that under this new restriction, a maximum inexpressible integer still always exists, and explore its values and properties as functions of $a,b$.

This paper initiates the study of arithmetic properties of closed points of arbitrary degree on curves. We show that a pointless curve of rank $0$ can have only finitely many cyclic degree $\ell$ points for all sufficiently large primes $\ell$. More generally, we show that the residue fields of points on curves of genus at least $2$ and rank $0$ are often biased away from total ramification. The proof technique illuminates a natural intermediate conjecture between the Manin–Mumford conjecture and the Zilber–Pink conjecture for self-products of abelian varieties. This intermediate conjecture implies analogous results for other splitting profiles.

The higher Green's functions $G_{k/2}(z,w)$ on the modular curve are real-valued eigenfunctions of the hyperbolic Laplacian in each variable, with logarithmic singularities along the diagonal. Gross and Zagier conjectured that their values at CM points, after multiplication by suitable powers of the discriminants, are logarithms of absolute values of algebraic numbers. The conjecture was proved by Li, Bruinier, and Yang. Mellit proposed a complex refinement of this conjecture, predicting that the normalized CM values of a complex lift of the Green function are logarithms of algebraic numbers modulo periods. For $\Gamma=\mathrm{SL}_2(\mathbb Z)$ and $k\in{4,6,8,10,14}$, we follow Zagier's construction of CM invariants from Eichler integrals of the Eisenstein series $E_4$. By applying Maass raising operators in both variables to a normalized iterated Eichler integral of a Bergman-type kernel, we obtain a complex-valued extension $\mathcal G_{k/2}^\bullet(z,w)$ whose real part is a constant multiple of $G_{k/2}(z,w)$. We study the normalized CM values of $\mathcal G_{k/2}^\bullet$ and describe their dependence on the choice of representatives of quadratic form classes. In particular, we show that replacing the quadratic form corresponding to the first variable by an equivalent one changes the exponential of the normalized value only by a root of unity. We also prove the algebraicity of these exponentials in the special case where the second CM point is $w_*=(1+i\sqrt d)/2$, with $d>3$ and $d\equiv3\pmod4$, and the class of the quadratic form corresponding to the first variable is equal to its inverse.

We introduce a systematic framework for constructing holonomic recurrences from meromorphic modular forms. Its central ingredient is a modular analogue of creative telescoping, which turns finite-dimensional quotients of spaces of meromorphic modular forms into explicit recurrence operators. This provides a common explanation for several previously isolated examples: it recovers Apéry's recurrence, Beukers' modular constructions and Zagier's seven sporadic recurrences. The framework also produces a large collection of new examples across different weights and groups. The modular origin of these sequences also confers them with non-trivial arithmetic properties, such as denominator bounds, irrationality statements, and $p$-adic congruences.

Let $X$ be a smooth projective variety over the complex numbers, and let $\mathbb{V}$ be an irreducible privileged $\mathrm{SL}_3$-local system on $X$ in the sense of Esnault and Groechenig. We show that $\mathbb{V}$ is of geometric origin. We also give an example of a privileged but non-rigid $\mathrm{SL}_3$-local system.

Let $K/\mathbb{Q}_p$ be a finite extension, and let $X/K$ be an algebraic variety. We present a new proof that the unipotent completion of the étale fundamental group of $X_{\bar{K}}$ is a pro-de Rham representation, and show that it is pro-semi-stable whenever $X$ has semi-stable étale cohomology. We also reprove, using syntomic methods, that it is pro-crystalline at integral base points whenever $X$ admits a smooth proper formal model. All three proofs use the bar construction and are inspired by Wojtkowiak's approach to unipotent fundamental groups.

Let lambda(n) denote the infimum of -min_x sum_{a in A} cos(ax) over all sets A of n positive integers. While the classical Sidon-difference construction yields lambda(n) <= (1 + o(1))sqrt(n)/2, we construct sets achieving a leading constant of 1/2: lambda(n) <= (1/2)sqrt(n) + O(n^(1/4)sqrt(log n)), establishing limsup_{n->infinity} lambda(n)/sqrt(n) <= 1/2. This construction combines a two-sided Sidon square identity, a Singer difference set, a Fejer-kernel majorant, a parity-biased deletion, and an exact-cardinality probabilistic selection. We also provide an L^4 criterion and a multiplicity bound constraining any construction with a smaller leading constant. In the opposite direction, we incorporate the L^1 norm into Bedert's boundary estimate with an explicit constant, yielding two lower bounds for f_A(x) = sum_{a in A} cos(ax): (1) -min f_A >= cn^(1/4) for every n-element set A subset of [1, Cn]; and (2) (-min f_A)^3 ||f_A||_1^2 >= cn for every n-element set A subset of N. The second bound recovers the 1/5 exponent of Bedert and Shankar and improves it whenever ||f_A||_1 is small relative to -min f_A. Numerical values of the construction are reported up to |A| = 25,440.

We prove a one-sided central limit theorem for $\log |L(\tfrac 12,\chi)|$ in a thin family of $n$-th order Hecke characters, for $n \geq 3$. The proof combines the method of Radziwiłł and Soundararajan with a recent asymptotic formula of Diaconu, Ion, Paşol, and Popa for the twisted second moment with a power-saving error term. We also obtain upper bounds of the conjectured order of magnitude for the $2k$-th moments for $0< k< 1$.

Periodically driven quantum systems exhibit a wide range of complex dynamics, including chaos. Yet, at special parameter values, their evolution can recur exactly. Several quantum protocols have recently been proposed that exploit exact Floquet recurrences to demonstrate genuine quantum advantage. Such recurrences can be studied through the arithmetic structure of the Floquet unitary. In this work, we establish a necessary and sufficient arithmetic criterion for exact recurrence and determine the minimal recurrence period when one exists. Because recurrence is a global property of the unitary, it holds for every initial state. Applying our framework requires only knowledge of the number field associated with the Hamiltonian parameters. We demonstrate the method using the kicked Ising chain, establishing exact recurrences in both Clifford and non-Clifford regimes. We further show that, in the recurrent non-Clifford regime, magic–a resource for quantum computation–is generated at intermediate times and undergoes exact periodic revivals enforced by the recurrence of the full Floquet unitary. By connecting number theory, quantum many-body dynamics, and quantum information, our work shows that exact recurrence extends beyond Clifford circuits and opens new avenues for magic generation and quantum hardware benchmarking.

We determine thirteen entirely CM identity fibers of Shimura parametrizations attached to elliptic curves of rank two and conductor below $2000$. Each yields a proof of the imaginary quadratic class number one theorem. The argument combines the Gross–Zagier formula with the Ribet–Takahashi degree comparison to show that the fiber is completely explained by known CM points.

We show that microlocal Arthur packets can be reconstructed from an otherwise underdetermined system of linear equations arising from Kashiwara's local index theorem. The key is a geometric rigidity property of Arthur orbits which, together with Fourier symmetry, uniquely determines the microlocal multiplicities defining these packets. This realizes the associated Arthur distribution as the unique normalized element in the intersection of two filtered spaces of standard stable distributions related by Fourier transform. Under standard enhanced local Langlands hypotheses and Aubert compatibility, this gives an alternative representation-theoretic definition of microlocal Arthur packets for general connected reductive $p$-adic groups that does not use twisted endoscopy and ensures stability through the stability of tempered packet distributions. In the unipotent setting, we use this characterization to prove agreement between endoscopic and microlocal Arthur packets for split $\mathrm{SO}_{2n+1}$ when the residue characteristic is at least $6n+4$. We further use this characterization to compute all unipotent Arthur packets with unramified real infinitesimal character for inner-to-split adjoint exceptional $p$-adic groups. In this exceptional setting, we prove the microlocal weak Arthur packet conjecture and give counterexamples to the microlocal analogues of the least- and greatest-parameter conjectures.

We construct a meromorphic Wilf–Zeilberger pair for the hypergeometric series for \(3\ln 2\), completing the WZ proof left open by Guillera and Rogers. The same pair yields a one-parameter identity and, after a half-integer shift and differentiation, an inverse-binomial series for \(\zeta(3)\). We also study the finite truncations of the logarithmic series. For every prime \(p\ge 5\), we prove that the lower truncation is congruent to \(q_p(2)-1\) modulo \(p\), while the upper truncation is congruent to \(-\frac{7}{16}p^2B_{p-3}\) modulo \(p^3\). The proofs combine the WZ method with terminating hypergeometric identities, the Pfaff–Saalschütz identity, and classical congruences for harmonic sums.

Ball's weak cylinder conjecture asserts that a set of $p^2$ points in $\mathrm{AG}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9/2}$. More precisely, a non-cylindrical graph of degree $D\ge3$ satisfies $|\mathcal{N}_S|\le p+1-\lceil p/(D-1)\rceil$ under the same hypothesis. This bound is attained when $p\equiv1\pmod{D-1}$. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in $p$. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.

We study Galois coinvariants of the geometric étale fundamental group and étale cohomology of varieties over a finitely generated field $k$. We establish finiteness properties of these coinvariants for (possibly infinite) Galois extensions $L/k$ such that $\operatorname{Gal}(L/k)^{\mathrm{ab}}$ has finite exponent. As an application, we obtain a finiteness result for étale fundamental groups of arithmetic schemes over number fields.

We improve the Plesken–Souvignier algorithm for computing automorphism groups and isometries of positive definite integer quadratic lattices by exploiting characteristic sublattices and refining vector enumeration. For lattices with roots, we also use the Weyl group to accelerate the computation of canonical forms. Experiments with concrete implementations demonstrate substantial speedups and show that canonical forms can be computed at a cost comparable to that of computing automorphism groups.

The classical Hurwitz bound gives a linear upper bound for automorphism groups of algebraic function fields in characteristic zero, whereas algebraic function fields in positive characteristic can have much larger automorphism groups. In this article, we firstly investigate solvable subgroups and $p$-subgroups of $\operatorname{Aut}(F/\mathbb{F}_q)$ where $p=\operatorname{char}(\mathbb{F}_q)$ and $F/\mathbb{F}_q$ is a function field of genus $g_F\ge2$ with full constant field $\mathbb{F}_q$. We prove that there are explicit constants $C_q^{(\mathrm{sol})}$ and $C_q^{(p)}$, depending only on $q$, such that, for every solvable subgroup $G\le\operatorname{Aut}(F/\mathbb{F}_q)$ and every $p$-subgroup $P\le\operatorname{Aut}(F/\mathbb{F}_q)$, \[ |G|\le C_q^{(\mathrm{sol})}(g_F-1), \qquad |P|\le C_q^{(p)}(g_F-1). \] In particular, one may take $C_q^{(\mathrm{sol})}=12q^2(q-1)$ for $q\ge4$ and $C_q^{(p)}=5q$ for every $q$. In the second part of this article , we give asymptotic lower bounds for solvable automorphism groups. For every $q$ there is an explicit constant $L_q>0$ and a sequence of function fields $F_i/\mathbb{F}_q$ with $g_{F_i}\to\infty$, together with solvable subgroups $G_i\le\operatorname{Aut}(F_i/\mathbb{F}_q)$, such that \[ |G_i|\ge L_q(g_{F_i}-1). \] For $q\ge5$ one may take $L_q=2q(q-1)/(q-2)$. If $q=r^2$ with $r\ge3$, an unramified tower construction over the Hermitian curve gives the stronger choice $L_q=2r^3(r-1)/(r-2)$. Moreover, as an application of our tower construction above, we improve the asymptotic lower bound of Ma and Xing \cite{MaXing2019} for subgroups of $\operatorname{Aut}(F/\mathbb{F}_q)$ whose order is relatively prime to $p$. As another application, we extend our method for bounding solvable automorphism groups to $p$-solvable subgroups, obtaining explicit linear bounds in terms of the genus.

We prove an upper bound for the error term in the higher rank hyperbolic lattice counting problem for ${\hbox{SL}(n, \mathbb{Z})}$. Blomer and Lutsko achieved recently a significant improvement of previous known results by Duke–Rudnick–Sarnak and Gorodnik–Nevo–Yehoshua. We deduce an improved upper bound for the error which holds for almost all pairs $z,w \in {\hbox{SL}(n, \mathbb{R})}$. We also derive a $\Omega$-result for the error term, in the spirit of Rudnick–Phillips.

We give a higher unit-torus lift from q-linearized permutation polynomials over Fqn to generally nonlinear sparse permutation polynomials over Fqns. The construction preserves compositional inverses: if a linearized polynomials and its inverse have coefficients (ai) and (Ai), then the lifted polynomial and its inverse have the same coefficients. As a consequence, we obtain nonlinear binomial lifts whose inverses are again binomials in the two-dimensional case, including an explicit inverse for a binomial family over Fq4. The three dimensional specialization yields permutation trinomials whose compositional inverses are again trinomials, together with explicit involution criteria and examples.

A set $S$ of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in $S$. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood \(B\) of \(0\) and every \(\eps>0\), restricting the common difference to \(B\) increases the extremal number for \(k\)-term arithmetic progressions by at most a factor of \(1+\eps\), provided \(k\) is sufficiently large.

Let $\mathbf{E}$ be the ring of values taken at algebraic points by $E$-functions. The exponential function and the Bessel function $J_0$ are classical examples of $E$-functions. We prove that $\mathbf{E}$ is not a field. More precisely, we prove that it is neither noetherian nor factorial, and construct non-units of $\mathbf{E}$ explicity, including $e+1$, $J_0(1)$, and ${}_1F_1(a;b;1)$ for rational parameters $a$ and $b$ such that $0

In this paper, we study the maximal length of a minimal zero-sum sequence over $\left[\!\left[ -m,m \right]\!\right]^d$ having a support of size $d+1$. When $d=2$, we find the exact value of this maximal length for all $m \ge 2$, as well as the exact structure of all extremal sequences for this problem. When $d=3$, we obtain lower and upper bounds that asymptotically match each other as $m$ tends to infinity. This allows us to improve on the best known lower bounds for the Davenport constant of low-dimensional boxes.

We study the least periods and multiplier orders in two constructions of Mercat for continued fractions with bounded partial quotients. Our main explicit result concerns a family in $\Q(\sqrt{127})$ with partial quotients at most $4$. For every integer $j\geq1$, we prove that its primitive period has length $92j+2$ and its multiplier order has conductor $y_{4j}/4$, where $(4730624+419775\sqrt{127})^k=x_k+y_k\sqrt{127}$. The totally positive unit group of this order has index $4j$ in that of the maximal order. These exact formulas give divisor-sum lower bounds at fixed traces and a lower bound of order $(\log X)^2$ for the number of bounded continued fractions with discriminant at most $X$. We also prove a primitive-period refinement of Mercat's palindromic construction. Combining this refinement with Hensley's counting theorem gives, for every $m\geq2$ and $\eta>0$, at least $X^{\delta_m/2-\eta}$ quadratic fields, up to a positive constant, that admit infinitely many purely periodic continued fractions with partial quotients at most $m$. Here $\delta_m$ is the Hausdorff dimension of the corresponding continued fraction Cantor set.