Papers

Let $F$ be an absolutely unramified mixed characteristic local field with rings of integers $O_F$. For a small affine algebra $R$ over $O_F$, we consider the logarithmic étale fundamental group $G_R$ of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara–Kato. We study $p$-adic representations of $G_R$ and obtain a classification of these representations in terms of étale $(\varphi, \Gamma_R)$-modules. Moreover, we define and study the notion of log-crystalline representations of $G_R$, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of $G_R$ are equivalent to Wach modules for $R$, extending our previous results from the (non-logarithmic) crystalline case.

Let $X$ be a smooth scheme of dimension $d$ over a field $F$. We study the abelian-group structure of the higher Chow groups $CH^{d+i}(X,j)$. For a prime $l$ different from the characteristic of $F$, we prove divisibility and torsion-freeness results when $i\ge$ the $l$-cohomological dimension of $F$. If $X$ is smooth proper and geometrically irreducible, we also study the kernel of the push-forward $CH^{d+i}(X,j)\to CH^i(F,j)$. We apply these results to finite fields, local fields, and global fields.

Recently, Li obtained a mean Lindelöf estimate for the cubic moment of the central values of Maass form symmetric square $L$-functions over short intervals $(T-H, T+H)$ of length $H \ge T^{18/19+\epsilon}$. We improve this result by showing that the estimate holds for $H \gg T^{6/7+\epsilon}$. The key ingredient in our proof is a new asymptotic formula for the second twisted moment of Maass symmetric square $L$-functions. Based on this formula, we also improve the lower bound for the proportion of non-vanishing central $L$-values in short intervals. Previously, even under the assumption of the Lindelöf hypothesis for Dirichlet $L$-functions, the proportion of non-vanishing values in intervals of length $H = T^{\beta}$ was only known to be at least $\frac{3\beta-1}{4}$. We establish an unconditional lower bound of $\frac{7\beta-2}{8}$.

We prove that every de Rham $p$-adic local system on a smooth projective curve over a $p$-adic field is potentially semistable; that is, it becomes semistable after pulling back along a finite cover of the curve. This establishes a relative version of the classical $p$-adic monodromy theorem of Berger and André–Kedlaya–Mebkhout. Along the way, we show that every $p$-adic differential equation near a type I\!V point on a curve becomes trivial after a finite étale extension.

In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called Nörlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the case of logarithm, we will show by the WZ method that for each order $\ell>1$ some Gregory coefficient of order $\ell$ must vanish, in contrast to the fact that all classical Gregory coefficients are nonzero. We also prove in this higher order logarithm case that the sequence is eventually alternating for each fixed order, a property enjoyed by the classical Gregory coefficients. In the most general setting, we conjecture that these new sequences are all eventually positive, which is supported by strong numerical evidence. Finally, we confirm this conjecture in the special case of polylogarithms and double polylogarithms. As a by product, for each zeta value and double zeta value, we find an infinite family of identities expressing its reciprocal as a sum of a rational number and an improper integral.

Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\sqrt{\log R\log\log R}$. We remove the factor $\sqrt{\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin–Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni–Zhang's conjecture on the nonvanishing of local relative characters.

For any primes $p \neq \ell$, $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_{p^2}$ have a typical structure called $\ell$-volcanoes, and the structure is the core of Sutherland's supersingularity testing algorithm for elliptic curves. In this paper, by exploiting the properties of $3$-isogenies between twisted Hessian curves, we show that when $p \equiv 2 \pmod{3}$ and $\ell = 3$, every ordinary twisted Hessian curve defined over $\mathbb{F}_p$ lies on the surface of the $3$-volcano. As an application, we give an improved version of Sutherland's supersingularity testing algorithm specialized to twisted Hessian curves defined over $\mathbb{F}_p$ with $p \equiv 2 \pmod{3}$. We also give a generalization of the known fact that any supersingular $j$-invariant is a cube in $\mathbb{F}_{p^2}$; we show that for any twisted Hessian curve $H(a,d)$ defined over $\mathbb{F}_{p^2}$, its $j$-invariant is not a cube in $\mathbb{F}_{p^2}$ if and only if $H(a,d)$ is ordinary and lies on the floor of a $3$-volcano.

The least interval diameter needed to realize every attainable cardinality of an $h$-fold sumset of a four-element integer set is $\binom{h+2}{2}+(h\bmod2)$ for $2\le h\le10$ and $d_h=(h-3)(h-1)$ for $h\ge11$. We prove the formula uniformly for $h\ge17$ and use complete finite searches, with a relation-rank coverage argument, for the remaining orders. For the lower bound, we show that every integer four-set with $\binom{h+3}{3}-17$ distinct $h$-fold sums has normalized diameter in $\{d_h,d_h+1,d_h+2\}$ when $h\ge9$, and we construct sets attaining $d_h$. For the upper bound, a relation-lattice index argument restricts larger minimum-diameter representatives to two or three active directions. Two general constructions and four special defects then provide short sets realizing every remaining cardinality.

Recent work of the author investigates certain random processes, valued in abelian p-groups, that naturally arise in the study of Haar random matrices over $\mathbb{Z}_p$. It was found, somewhat surprisingly, that these processes are reversible Markov chains. In this short note, we give a simple alternative derivation of this fact. The new derivation also proves that this phenomenon is not specific to $\mathbb{Z}_p$, but generalizes to Haar random matrices over any profinite local ring.

The solution of the Diophantine equation ay^2=b(x^k-c^k) is studied in different cases of k. Methods are proposed for different cases of k, which are convenient for computer algorithm implementation. The original equation is reduced to lower-degree equations, which can improve the efficiency in searching possible integer solutions. A sufficient condition of the unsolvability of ay^2=b(x^2n-c^2n) is presented.

We compare return probabilities for random walks stopped at an independent geometric time on Lubotzky–Phillips–Sarnak graphs and on the infinite regular tree, with the McKay measure giving the tree value and the Ihara zeta function describing the additional returns caused by cycles. We estimate this correction when the mean walk length is comparable to the length of the shortest cycle, a regime in which walks long enough to traverse a cycle receive non-negligible weight. Unconditionally, we prove an upper bound with a double logarithmic factor and bounded fixed moments over prime levels after multiplication by the number of vertices. Under the Generalized Riemann Hypothesis for quadratic Dirichlet $L$-functions, the correction is of the same order as the reciprocal of the number of vertices.

For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le(2\log2+o(1))\,n/\log n$. This improves the bounds of order $n/(\log n)^{1/2-\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above $\sqrt{n!}$, with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant $2\log2$.

Using estimates established in a previous paper, we prove quantitative universality results for cokernels of products of i.i.d. random matrices and for flags associated to k-tuples of random i.i.d. matrices, over finite local rings. In the case when the ring is a quotient of $\mathbb{Z}_p$, this gives a quantitative analogue of some previous results of Huang, Nguyen and Van Peski.

We consider the cokernel corners process for an i.i.d. matrix with entries in a finite local ring. When the distribution of the entries is uniform, this process is a Markov chain, and hence the ergodic theorem for Markov chains can be applied. This implies, in particular, that for uniformly distributed p-adic random matrices, the cokernels of the corners are distributed according to the Cohen-Lenstra measure, almost surely. The purpose of this note is to show that the conclusion of the ergodic theorem also holds for i.i.d matrices, provided that the distribution of the entries is not concentrated on the translate of a subring, or the translate of an ideal. This will follow from the bounds proved in a previous paper of the author.

We identify the precise F-theory backgrounds dual to the heterotic string theory compactified on $T^2$ with general Wilson lines turned on for one of the two $E_8$'s. The elliptic curve describing the backgrounds is expressed in terms of orthogonal modular forms and is determined so that it admits a nontrivial scaling limit yielding a rational elliptic surface. We confirm that the curve reduces to the Seiberg-Witten curve for the E-string theory in this limit. We conjecture that our result gives an explicit inverse period map for algebraic K3 surfaces polarized by the $U\oplus E_8(-1)$ lattice.

We describe a new greatest common divisor (GCD) algorithm for polynomials with integer coefficients. The bit complexity of the new algorithm is polynomial in the input and output sizes and the individual degree bounds.Our algorithm follows the standard approach by reducing multivariate polynomial GCD to univariate polynomial GCD. Our algorithm reduces a multivariate polynomial GCD to a single univariate polynomial GCD. The main idea of our algorithm is a new variable substitution which reduces a multivariate polynomial to a separated one, that is, the coefficients in a main variable are all monomials. The explicit bit complexity is analyzed and we have implemented our algorithm in Maple. It is shown that our algorithm is efficient for polynomials with high degree, large number of variables, but small number of terms in GCD.

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ \nu_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ \nu_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino–sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.

We prove that, for every positive integer $n$, the polynomial $C_0+C_1x+\cdots+C_nx^n$ is irreducible over $\mathbb{Q}$, where $C_k$ denotes the $k$th Catalan number.

We characterize the standard modules of $\GL_{2m}(\C)$ that are distinguished by $\GL_m(\HH)$. Let $\delta_1,\dots, \delta_{2m}$ be characters of $\mathbb{C}^\times$. Assume that $S = \delta_1 \times \cdots \times \delta_{2m}$ is a standard module of $\GL_{2m}(\mathbb{C})$. For each $i$, define ${\delta_i^*} (z) = \delta_i(\overline{z})^{-1}$ for $ z \in \mathbb{C}^\times$. In particular, we conclude that a standard module for $\GL_{2m}(\mathbb{C})$ is distinguished by $\GL_{m}(\mathbb{H})$ if and only if there exists an involution $p\in S_{2m}$ without fixed points such that $\delta_{p(i)}=\delta_i^*$ for every $i$. We first verify the hypotheses of the multiplicity estimate theorem of Suzuki and Tamori in \cite{ST}. The orbit calculation of Matringe, Offen, and Yang in \cite{MOYglobal} then gives the necessary condition and a dimension bound. Then local intertwining periods prove sufficiency.

We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke–Maass cusp forms for the group $\mathrm{SL}_n(\mathbb Z)$, where $n\geq 2$ is an integer. Our main result establishes, under the Generalised Ramanujan–Petersson Conjecture, that for all $X\leq x\leq 2X$ outside a set of measure $o(X)$ the short interval $(x,x+H]$ contains at least $H(\log X)^{1/n^2-1}$ indices with both positive and negative Hecke eigenvalues, provided that $(\log X)^{1-1/n^2+\varepsilon}\leq H\leq X/10$. For $\mathrm{GL}_2$ Hecke–Maass cusp forms we obtain a stronger bound unconditionally, and for self-dual $\mathrm{GL}_3$ forms we obtain the same logarithmic exponent unconditionally. We also establish results under weaker bounds towards the Generalised Ramanujan–Petersson Conjecture together with the Generalised Lindelöf Hypothesis, and obtain lower bounds for the number of sign changes and for the number of Hecke eigenvalues of each sign in long intervals.

We determine the integer group determinant sets for $C_{24}$, $C_2\times C_{12}$ and $C_2^2\times C_6$, the three abelian groups of order $24$. Character factorization expresses each determinant as a product of cyclotomic norms, whose components must satisfy integral compatibility conditions. The classifications are constructive and reduce each nonzero signed integer to valuation conditions at $2$ and $3$ and a finite test on its remaining prime factors. The tests use conductor quotients and norm-preserving global units. Each prime contribution has a uniform bound independent of its multiplicity, and successful tests yield integer coefficients realizing the prescribed determinant. The critical cyclic strata share one compatibility group. For $C_2^2\times C_6$, a normalized rational–Eisenstein pair settles the odd values and most even strata; target stabilizers reduce the remaining tests. We also prove that the odd determinant set of $C_2^2\times C_6$ is properly contained in that of $C_2\times C_{12}$, exhibit an infinite family in the difference, and determine the signed powers of two in all three sets. Exact arithmetic certificates and coefficient constructors accompany the proofs.

We introduce a new type of $p$-adic hypergeometric function, which satisfies congruence relations similar to Dwork's $p$-adic hypergeometric function. Also, we prove that the syntomic regulators of the Hesse cubic curves are expressed in terms of the special values of our new $p$-adic hypergeometric functions. We also show that there is a transformation formula between our new $p$-adic hypergeometric function and a $p$-adic hypergeometric function of logarithmic type defined by Asakura.

We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite–Minkowski theorem. The proof establishes strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\arcsin x)/x$ on $(-1,1)$. A positive-kernel formula and the classical Schur algorithm give these signs, and a para-orthogonal transformation yields positivity in every degree. We also compute the positive density representing the negative of the Schur function as a Hausdorff moment generating function. After rescaling their indices to $[0,1]$, the normalized critical vectors converge weakly to the arcsine law, while Chenevier's critical scale is asymptotic to $8\pi/n$. At the critical boundary, a single nonzero effective integral vector is negative for every admissible test function exactly in odd degree and in degree zero. Finally, we prove an exact first-variation formula for exponential perturbations of the Legendre measure and derive its asymptotics for endpoint cusps. For the perturbation leading to Chenevier's weight, the derivative at the Legendre measure has a $(\log n)/(\pi^2n^2)$ term and an explicit constant at order $n^{-2}$. The corresponding nonlinear asymptotic remains conjectural.

We establish sharp local estimates for smooth numbers. Corollaries are derived, including optimal estimates for square-free smooth numbers.

We develop an algebraic framework for the quasi-modularity of symmetric multiple $q$-zeta values. We identify natural classes of symmetric quasi-shuffles whose $q$-zeta values exhaust the algebra of level-one quasi-modular forms, and further classes whose $q$-zeta values are quasi-modular forms of finite level. We also obtain explicit symmetrization formulas and relate symmetric quasi-shuffles to symmetric and quasisymmetric functions.

We give a new proof of Faltings' isogeny theorem. More precisely, we show that Yuan's non-archimedean equidistribution theorem can be used to "pump" homomorphisms and semisimplicity from finite fields to number fields, thereby reducing Faltings' theorem directly to Tate's theorem. In the appendix joint with Yap, we explain how an ultraproduct variant of our method gives another proof of the Shafarevich conjecture, and hence a new route to the Mordell conjecture via Parshin's trick.

We prove upper bounds for certain number field counting functions using Serre's modularity conjecture (now a theorem of Khare–Wintenberger). These results are comparable to sharp upper bounds for the number of abelian extensions with fixed or bounded discriminant, with Serre's modularity conjecture playing the role of class field theory.

Let $p$ be a prime and let $A,B$ be nonempty subsets of the cyclic group $\mathbb{Z}_p$ with $|A|\neq |B|$. The Alon–Nathanson–Ruzsa theorem gives the lower bound $|A\rplus B|\ge \min\{p,\,|A|+|B|-2\},$ where $A\rplus B=\{a+b:a\in A,\ b\in B,\ a\neq b\}$ is the restricted sumset. The inverse problem of characterizing all critical pairs $(A,B)$ attaining equality was posed by Alon, Nathanson, and Ruzsa in 1996 and remains open. Recently, Liu and Qian solved the inverse problem under the assumption that at least one of the sets is an arithmetic progression, and proposed a conjecture for the general case. The main purpose of this paper is to show that this conjecture fails in the boundary case $|A|+|B|=p$. More precisely, we construct infinite families of non-arithmetic critical pairs $(A,B)$ with $|A|+|B|=p$ and $|A\rplus B|=p-2$ for every prime $p\ge 11$. These families show that the boundary case is fundamentally different from the non-boundary case. Motivated by this, we formulate and prove a refined inverse theorem under the natural hypothesis $|A|+|B|\le p-1$, which excludes the boundary.

Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, $\beta \in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ \Delta \in (0, 1/2)$. For any real $y$, let $\|y\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely many primes $\ell \equiv u \bmod v$ such that $ \|\alpha \ell - \beta\| \ll_v \ell^{-1/4} \log^{8} \ell. $ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary. Before stating the result, let us define a function on $f_{\Delta}(\theta)$ on $\R$ such that $f_{\Delta}(\theta)$ is $ 1 \text{ if } \| \theta \| < \Delta$ and $ 0 $ otherwise. Further, suppose that $u, v \in \mathbb{Z}_{\ge 1}$ are coprime and $a$, $q \in \Z$ are coprime with $q> N^{1/4}$ and $|\alpha v -a/q| \le 1/q^2$. Then, for every $\epsilon \in \mathbb{R}_{>0}$ we have \begin{equation*} \sum_{\substack{n=1 \\ n \equiv u \bmod v}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}\Delta^{1/2} + (\Delta Nq)^{1/2} + N^{\epsilon}q \Delta^{1-\epsilon}) \mathcal{L}^8 \end{equation*} where $\mathcal{L}=\log (Nq/\Delta)$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta). \end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \subset \R\setminus \Q$ such that for every $\gamma \in S$ there are infinitely many primes $\ell \equiv u \bmod v$ satisfying $\| \gamma \ell \| < \ell^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erdös and Mahler from 1939.