Papers

In this paper, we present a one-parameter extension of some supercongruences related to Van Hamme's $\mathrm{(E.2)}$ and $\mathrm{(F.2)}$ supercongruences. To deduce these results, we first establish the corresponding $q$-supercongruences modulo the third power of a cyclotomic polynomial. Our proofs rely on the creative microscoping method devised by Guo and Zudilin~\cite{guo2019q}, together with Jackson's ${}_{6}\phi_{5}$ summation formula.

In this paper, we establish several $q$-supercongruences for basic hypergeometric series modulo the third and fifth powers of a cyclotomic polynomial. These results extend several existing supercongruences related to the $\mathrm{(A.2)}$ and $\mathrm{(C.2)}$ supercongruences of Van Hamme. Our proofs employ the creative microscoping method developed by Guo and Zudilin \cite{guo2019q}, together with Watson's ${}_{8}\phi_7$ transformation formula.

The study of the distribution of real sequences in the unit torus has a long and rich history. There are many notions which describe the “pseudo-random” behavior of a given deterministic sequence (often of arithmetic origin), such as equidistribution (which describes the distribution on a “global scale”) or the pair correlation (which is a “local-scale” statistics). Another popular and important statistics, which can be applied to all ranges from global via intermediate to local, is the number variance $V_N(s)$, which describes the mean-square fluctuation of the number of elements of an initial segment of the sequence in an interval of length $s$, when moving this interval around the torus. In the terminology of quantum chaology, where such statistics are used to describe the distribution of the energy levels of a system, pseudo-random behavior is called Poissonian behavior. Since the number variance is an important indicator to qualify pseudo-random behavior, it is quite surprising that the literature does not seem to contain a precise description of the distribution and of the almost sure asymptotic behavior of the number variance in the truly random case (i.i.d.\ random points), even in the comparatively simple setup of uniformly distributed points on the unit torus. In the present paper we calculate the distribution of the number variance with high precision throughout a wide range of the parameter $s$, and determine the precise threshold (in terms of $s$) where the “Poissonian” asymptotics $V_N(s) \sim N s$ breaks for a typical realization of a random sequence. Key technical input comes from the Komlós–Major–Tusnády theorem and the Karhunen-Loève decomposition.

In this paper, we treat the problem of determining when a sequence $\{c_n\}_{n=0}^{\infty}$ with the exponential generating function $F(t)$ satisfies the congruence $c_{n+(p^h-1)p^k}\equiv c_n \pmod{p^{k+r}}$ for large $n$ and when its canonical interpolating functions are locally analytic. This type of congruence is often observed for several number-theoretic or combinatorial sequences, such as Bernoulli numbers, Euler numbers, and Fibonacci numbers. Traditionally, such congruences are proved by using $p$-adic integrations. In this paper, we present another approach to this problem. Instead of using $p$-adic integrals, we investigate an algebraic structure of the set of generating functions and establish some conditions on $F(t)$ that are equivalent to or sufficient for the congruence and smoothness of the interpolating functions. We also provide concrete applications of our framework to Genocchi numbers, generalized Euler numbers, and so on, which are outside the scope of the typical $p$-adic integration method.

Let $k$ be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over $k$ and over $\mathbb{C}$ equipped with the hybrid norm. In both settings, we show an analogue of Mané–Sad–Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.

We introduce multiple $s$ values and prove that every alternating multiple zeta value is a $\mathbb{Q}$-linear combination of multiple $s$ values $s(k_1, \dots, k_r)$ with $k_i \in \{1, 2\}$. We also study when certain multiple $s$ values can be expressed as $\mathbb{Q}$-linear combinations of multiple zeta values.

We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequency over short intervals. As an application, we prove that \[ \Big|\sum_{x<n \leq x+H}\mu(n) e(n\alpha)\Big| \ll H(\log x)^{-1/3+\varepsilon } \] whenever $x^{0.58+\varepsilon }\leq H\leq x$. In the same range of $H$ we also show that \[ \Big|\sum_{x<n \leq x+H} \Lambda (n) e(n\alpha)\Big| \ll H(\log x)^{-A} \] for arbitrarily large $A>0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\|q\alpha \| \ll x (\log x)^{O_A(1)}/H^2$. This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials would lead to a corresponding improvement in the lower bound for $H$.

We construct a positive, finitely additive, left-invariant measure on the congruence cylinder ring of a smooth $d$-dimensional algebraic group over an $n$-dimensional local field. The measure takes values in $\mathbb{R}((X_2))\cdots((X_n))$. A normalized cylinder of depth $(i_1,\ldots,i_n)$ has volume $q^{-di_1}\prod_{\ell=2}^nX_\ell^{di_\ell}$, where $q$ is the final residue-field cardinality. We also obtain coordinate-change and quotient integration formulas for this measure.

We introduce weak $A_n$-formality for Galois cochain algebras and, for profinite groups whose mod-$p$ cohomology is concentrated in degrees at most two, characterise weak and genuine $A_n$-formality through finite embedding problems and the projective system they form. We then study $A_n$-formality for number fields. Let $K$ be a number field and let $p$ be an odd prime with $\mu_p\not\subset K$. For $p\geq 5$, and for $p=3$ under a natural local cyclotomic condition, we prove that every associated finite $A_3$-embedding problem is solvable. In contrast, for every odd $p$ with $\mu_p\not\subset K$, we show that no compatible family of finite-stage solutions exists, and hence $C^\bullet(G_K,\mathbb F_p)$ is not genuinely $A_3$-formal. The proof uses Chebotarev density, governing fields, and the Gras-Munnier criterion to obstruct the compatibility of finite-stage solutions, and it builds on previous work of Maire-Mináč-Ramakrishna-Tân. This shows that genuine $A_3$-formality detects a global compatibility obstruction for number fields invisible to every individual finite embedding problem and invisible to obstructions arising from individual strong Massey vanishing problems.

We define and study six ray class monoids associated to a level datum $(\mathcal{O}; \mathfrak{m}, \Sigma)$, where $\mathcal{O}$ is an order in a number field $K$, $\mathfrak{m}$ is an $\mathcal{O}$-ideal, and $\Sigma$ is a set of real places of $K$. The monoids extend the ray class group (and the ring class group) by including non-invertible classes consisting of non-invertible ideals and ideals not coprime to the modulus. We prove fundamental results on the structure of these ray class monoids.

Extending recent works on the restricted divisor function over integers, we consider an analogue of this function over the function field. We provide some exact and asymptotic formulas related to this function.

We study non-vanishing modulo primes of critical values of $L$-functions over imaginary quadratic fields twisted by Coates–Wiles characters. For a parallel weight two Hecke eigenform over an imaginary quadratic field, we prove that a positive proportion of Coates–Wiles characters have nonzero integral $L$-values modulo each prime in a positive-density set. The argument constructs automorphic symbols in parabolic homology with integral coefficients and expresses the critical values through their pairings with parabolic cohomology classes. A vertical family of additive averages of these pairings recovers Fourier coefficients and force the module generated by symbols to have full rank. We transfer this full-rank property to non-vanishing modulo primes. This extends the homological strategy of Kim–Sun from classical modular curves to arithmetic orbifolds, while addressing the unit obstructions specific to the imaginary quadratic setting.

Denote by $\mathrm{Kl}(a,n)$ the normalized Kloosterman sum modulo $n$. In this paper, we study the moments of $|\mathrm{Kl}(1,n)|$ on average over $n\leqslant X$, and show that \begin{align*} \sum_{n\leqslant X}|\mathrm{Kl}(1,n)|\asymp X(\log X)^{\frac{8}{3\pi}-1}. \end{align*} The lower bound is proven by considering a subfamily of moduli which factorize in a nice way such that vertical Sato–Tate distributions of Kloosterman sums (as variants of Katz) apply. The upper bound is based on a new density version of upper bound sieves, combining with averages of divisor functions in arithmetic progressions and a similar vertical Sato–Tate distribution.

We provide an elementary approach to computing the quadratic Gauss sum, emphasizing the method employed to compute the sum corresponding to multiples of 4. The proof method employed does not rely on the law of quadratic reciprocity, but only on exploiting symmetries among the sum's components, together with a recursive approach.

The real-axis jump of the prime-zeta function gives an integral form of Gram's series for Riemann's $R$-function. Applying the same operation to the classical almost-prime generating functions defines a smooth family $R_k$, with $R_1=R$. We give the construction and a convergence argument, and explain why its expansion at every fixed logarithmic order is the Selberg–Delange expansion. The distinction lies in the smaller real-axis contributions retained by the definition, rather than in new asymptotic coefficients. A finite Perron decomposition also identifies the additional contour terms required for exact recovery of the count.

In this paper, we prove a conditional result that yields an asymptotic formula with an explicit main term and a power-saving error term for the generalized shifted ternary additive divisor sum $D_{\mathcal{I},\mathcal{J}}(X,1)=\sum_{n \leq X} \tau_{\mathcal{I}}(n) \tau_{\mathcal{J}}(n + 1)$, where $\mathcal{I}=\{\alpha_1,\alpha_2,\alpha_3\}$ and $\mathcal{J} = \{\beta_1, \beta_2, \beta_3\}$, with $|\alpha_i|, |\beta_i|\leq \delta$ for $i=1,2,3$, and $0<\delta<1/44$. The proof of our main theorem relies on assuming an averaged level of distribution for the generalized ternary additive divisor function $\tau_{\mathcal{J}}$ in arithmetic progressions up to the level $2/3$.

Let $\Gamma<\mathrm{SU}(n,1)$, $n\ge2$, be Zariski-dense and contain a regular loxodromic element $A$. The splitting-field problem asks how far traces and spectra determine a field of definition for $\Gamma$. We show first that traces together with the spectrum of $A$ always give a linear realization over $K=\mathbb{Q}(\textrm{tr}\Gamma,E(A))$. We also prove that every complex-irreducible subgroup of $\mathrm{SU}(n,1)$ with real trace field is conjugate into $\mathrm{SO}(n,1)$; in particular, no discreteness hypothesis is required for this real-trace conclusion. In higher dimension, however, linear descent does not by itself give the standard unitary form: a Hermitian similarity obstruction remains, described over imaginary quadratic fields by Landherr's local invariants. In $\mathrm{SU}(3,1)$ we give a Zariski-dense example for which the one-spectrum field is $\mathbb{Q}(i)$ but does not yield the standard unitary form. This leads naturally to the total spectral field. We prove that the obstruction disappears for arithmetic groups and under suitable local openness hypotheses, and we give an even-degree local criterion. In dimension four we analyze the discrete branch through explicit Schottky groups and rank-one local geometry. There the full Bruhat cocycle recovers anisotropic Levi holonomy and forces a non-split rotational word. The remaining quasi-split rank-two case is not reached by this method, so the general $\mathrm{SU}(3,1)$ problem remains open.

We find the exact rate of equidistribution of several arithmetic point configurations in Wasserstein metric and $L^2$ discrepancy with respect to balls. Kronecker sequences in the torus and the Fibonacci lattice on the Euclidean unit sphere are shown to be more evenly distributed in Wasserstein metric than random points. Modular hyperbolas and modular parabolas in the torus behave like random points, although the constants in the asymptotics of the $L^2$ discrepancy with respect to balls differ from the random case. We also prove a sharp inequality between the quadratic Wasserstein metric and the $L^2$ discrepancy with respect to balls for an arbitrary probability measure on the torus or the sphere. In particular, we show that any point set that is optimally close to the uniform measure in $L^2$ discrepancy with respect to balls is also optimal in the quadratic Wasserstein metric.

Motivated by applications in Diophantine approximation, we develop a new fibering mechanism for non-degenerate manifolds in $\mathbb{R}^n$, showing that they admit local decompositions into families of non-degenerate curves with uniform quantitative non-degeneracy properties. The manifolds are required to be $C^N$ with the essentially optimal bound $N\geq \max\lbrace n,l+1\rbrace$, where the manifold is $l$ non-degenerate. Furthermore, we show that this uniform quantitative local control can be globalised into a qualitative statement, so that almost every point of a non-degenerate manifold lies on a non-degenerate curve contained in the manifold. Our approach is perturbative and shows that the fibering structure arising in the analytic setting is stable under sufficiently small $C^N$ perturbations. The method provides a framework for extending fibering based results in Diophantine approximation from curves to manifolds. Such applications are considered in the paper.

We develop a modular wall crossing theory for quasimaps to a projective variety, allowing independent variation of the boundary coefficients and the quasimap weight. Building on the K-stability of quasimaps introduced by Hashizume and the first author, we construct projective moduli spaces in the stable, Calabi–Yau, and log Fano regimes, together with wall crossing morphisms. A central construction is the moduli theory of boundary polarized Calabi–Yau quasimaps, which retains an ample polarization at the numerically trivial locus and allows comparison with suitable perturbations toward the stable and log Fano regions. The stable theory applies in arbitrary genus, while the comparisons through the Calabi–Yau locus concern genus zero. For degree-one boundary divisors, the resulting framework relates weighted stable maps and quasimaps to Hassett spaces and GIT quotients of weighted points on $\mathbb P^1$. In a companion paper, we apply this framework to give a modular interpolation between Miranda's GIT compactification of rational elliptic surfaces and the Baily–Borel compactification of an eight-dimensional ball quotient.

Using K-moduli spaces for log quasimaps $q_t\colon\left(\mathbb P^1,\frac{1-t}{12}D\right)\to[\mathbb A^2/\mathbb G_m]$ of degree twelve with twelve points and weight $t/12$, we construct a modular interpolation $\{\mathcal M_t\}_{0\le t\le1}$ between the Baily–Borel compactification of the Heckman–Looijenga ball quotient $X_o$ and Miranda's GIT compactification of the moduli space of rational elliptic surfaces. We completely determine the wall-crossing and, for every rational $t\in[0,1]$, identify \[ \mathcal M_t\cong\operatorname{Proj}R\!\left(X_o,\mathcal L+\frac t2\Delta(6)+\frac t3\Delta(9)\right), \] where $\mathcal L$ is the automorphic $\mathbb Q$-line bundle and $\Delta(6),\Delta(9)$ are distinguished Heegner divisors. On the automorphic side, we construct a new automorphic form on $X_o$ via a Borcherds product, whose divisor gives an independent relation among the Heegner divisors. This relation provides a key input for determining the birational transformations in the K-moduli wall-crossing. As part of this analysis, we show that the first positive chamber $\mathcal M_t$ for $t\in (0,1/7)$ is Looijenga's semi-toroidal compactification.

Let $U$ be the ring of upper triangular $2\times2$ integer matrices. For $N\in U$, a Jordan $D(N)$-set is a set of distinct nonzero matrices in $U$ such that $(AB+BA)/2+N$ is a square in $U$ for any two distinct elements $A$ and $B$. We determine all $N$ for which an infinite Jordan $D(N)$-set exists. If no infinite set exists, every such set has at most six elements. If $N$ is not a difference of two squares in $U$, the bound is five. Both bounds are best possible. We also give necessary and sufficient conditions for the existence of infinite sets with nonzero pairwise Jordan products and of infinite sets consisting of nonsingular matrices.

Let $f\in\mathbb Q[t]$ be an irreducible polynomial of positive degree with positive leading coefficient, taking integer values at every integer. We prove that there is a constant $c_f>0$ such that, for every sufficiently large real $X$, the interval $(X/2,X]$ contains at least $c_f\log X\log\log X$ consecutive integers $n$ for which all values $f(n)$ are composite. The exponent of $\log\log X$ is independent of the degree of $f$. This improves the previous result of Ford and Gabdullin by replacing their small fixed power of $\log\log X$ with the full factor $\log\log X$. In particular, the result applies to $f(t)=t^2+1$.

In this paper we propose and study what we call the Erdős-Falconer distance problem in the integer lattice. One motivation is Magyar's discrete analog of the result of Furstenberg, Katznelson, and Weiss, who showed that all arbitrarily large distances had to appear for a set of positive upper Banach density in $\mathbb{R}^d$, $d\geq 2$. Another motivation is the Falconer distance problem, which can be viewed as a refinement of a result of Steinhaus, dropping assumptions from positive Lebesgue measure to merely asking about the dimension of sets. In this context we thus want to lower the assumption of positive upper Banach density and yet obtain statements that guarantee that we have lots of distinct distances. All of these questions are also related to the Erdős distinct distance problem. Our naming is as for the finite field variant of the problem, acknowledging that parts of the problem share similarities with the Erdős distinct distance problem while others share similarities with the Falconer distance problem. The main tool we use are $\ell^p$-improving inequalities for discrete spherical averages, which to the best of our knowledge is the first application of such bounds. One can use results on the Erdős distinct distance problem to make progress on our conjecture but surprisingly, even if using the strongest available bounds in high dimensions using the mechanism of Solymosi and Vu, then in certain regimes our methods do better. While a resolution of the Erdős distinct distance problem would imply our unpinned conjecture then no results on either the Erdős or Falconer distance problem seem to imply any results for our pinned variant. We view this problem as a model problem that might ultimately contribute back to the original pinned distance problems.

We show that for a fixed $\mathrm{SL}_3(\mathbb Z)$ Hecke–Maass cusp form $F$, there are infinitely many primitive Dirichlet characters $\chi$ such that $L(1/2, F\times \chi)$ and $L(1/2, \chi)$ do not vanish simultaneously. More precisely, we obtain an asympotic formula with a power-saving error term for the product of these central values averaged over $\{\chi\bmod q: q\in \mathcal Q\}$ where $\mathcal Q$ is a set of squarefree integers with a small divisor of suitable size.

We study three quadratic Collatz-type maps on the non-negative integers. Each of them halves an even number, and they send an odd $n$ to $n(n-1)/2$, to $n(n+1)/2$ and to $(n^2 - 1)/4$, respectively. Along an orbit, the odd terms grow until they reach a number of the form $2^m + 1$ or $2^m - 1$, depending on the map, and an orbit with an odd term greater than $1$ is bounded exactly when this happens. The step that reaches such a number gives a solution of an exponential Diophantine equation, which becomes the equation $2^a \pm 2^b + 1 = z^2$ after completing the square. With Szalay's classification of the solutions of this equation, we classify all orbits of the maps. Every orbit either is eventually periodic or tends to infinity, and we determine exactly the initial values with bounded orbits. For the map with odd branch $n(n-1)/2$, Sedaghat stated the classification, and we give a new and complete proof. For this map, we also give an elementary criterion for unbounded orbits, based on divisibility alone. To our knowledge, the classifications for the other two maps are new.

For a positive integer $n$, we study the irreducibility of the generalised Laguerre polynomials \[ L_n^{(a)}(x) = \frac{1}{n!}\sum_{j=0}^{n} \binom{n}{j} \frac{(n+a)!}{(j+a)!}(-1)^j x^j, \] where $a \in \mathbb{Z}_{\ge 0}$. Earlier works established that these polynomials are irreducible over $\mathbb{Q}$ for $0 \le a \le 50$. In this paper, we substantially strengthen these results by establishing that $L_n^{(a)}(x)$ are irreducible for all $a \le 1000$. We derive sharper constraints on potential integer roots and make systematic use of the geometry of the associated Newton polygons. This approach allows us to extend the irreducibility range significantly while keeping explicit computations to a minimum.

The discovery that the generalized Riemann functional equation from the literature can be extended to real $b \ge 0$, by exploiting symmetries of a new formula for the Hurwitz zeta function with respect to the shift parameter $(b)$, led to the search for a similar relation for the Lerch $\Phi$ function, $\Phi(e^z,k,b)$, where a new formula is used as a starting point and the symmetries with respect to $b$ are exploited again. This search resulted in a generalization of the Riemann functional equation with three variables, whose domain is then extended to $b>1$, with the aid of a new self-contained Lerch $\Phi$ formula, making the relation hold for all real $b$. As a secondary objective, we address the problem of obtaining the analytic continuation of a finite summation that appears in a formula for the Lerch $\Phi$ function. Above all, this paper demonstrates, with clear elementary mathematics, how a functional equation can be derived.

We study the structure of the $p$-class group of the first layer of the cyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field where $p$ is non-split. We obtain restrictions on the growth of the cyclic factors from the base layer to the first layer; in particular, when the $p$-rank remains unchanged, each cyclic factor grows by exactly one power of $p$. We show that in a large number of cases the Iwasawa $\lambda$-invariant is completely determined by the $p$-rank of the first layer itself. Finally we use a Cohen-Lenstra-Martinet type philosophy to study the probability distribution of Iwasawa lambda invariants of the first layer when the base layer has cyclic $p$-class group.

We extend the Nair–Tenenbaum estimates for short sums of nonnegative arithmetic functions at polynomial values to mixed moments involving Kloosterman sums. Combining these estimates with lower bounds obtained by refining the method of Fouvry and Michel, we prove that, for every fixed nonzero integer \(a\) and fixed \(z,\nu>0\), \[ \sum_{n\leq x}z^{\omega(n)} |\operatorname{Kl}(a;n)|^\nu \asymp_{a,z,\nu} x(\log x)^{z\mathfrak{s}(\nu)-1}, \] where \(\omega(n)\) counts distinct prime divisors and \(\mathfrak{s}(\nu)\) is the \(\nu\)-th absolute moment of the Sato–Tate measure. We also obtain upper bounds for mixed absolute moments of Hecke eigenvalues and Kloosterman sums, determine the frequency of large Kloosterman values on a logarithmic scale, and establish the order of magnitude predicted by Li and Sarnak for the diagonal second moment of classical Kloosterman sums. We obtain matching bounds for the smoothed spectral counting variance on the modular surface in a logarithmic range and the lower bound \(\int_T^{2T}S(t)^2\,dt\gg T^2/\log T\), where \(S(t)\) is the remainder in Weyl's law.