math.NT

Pair correlation of $αn^θ$ for random $θ$

For fixed $α>0$ , we show that the sequence $\{αn^θ\}$ has Poissonian pair correlation for Lebesgue-almost all $θ\in (0,\frac{3}{5})\cup(3,\infty)$ . This improves a result of Technau and Yesha, who proved the same for almost all $θ>7$ . The approach of Technau and Yesha was based on a repulsion principle, which roughly allows one to estimate the variance of the pair correlation function using the fourth derivative of the phase. In our approach, we split the $θ$ -integration in the variance into many short intervals and show that most of the integrals can be estimated using the first derivative. The problem is then reduced to several counting estimates, which we prove using moments of the Riemann zeta function and exponent pairs.

A $q$ -Analogue of a Supercongruence Related to Van Hamme's (B.2) Supercongruence

Motivated by the recent work of Li and Wang on parametric generalizations of Van Hamme's $(C.2)$ supercongruence in the $q$ -setting, we establish $q$ -analogues of a supercongruence related to Van Hamme's $(B.2)$ supercongruence, recently obtained by the authors. In particular, we derive parametric extensions of these $q$ -supercongruences by constructing suitable pairs of hypergeometric functions through the $q$ -WZ method.

The Abel--Jacobi map over the twistor- $\mathbb{P}^1$ and real local class field theory

We study the Abel--Jacobi map over the twistor- $\mathbb{P}^1$ in the context of Scholze's geometrisation of the real local Langlands correspondence. In a similar spirit to a result of Fargues over the Fargues--Fontaine curve, we prove that pullback along the Abel--Jacobi map induces an equivalence on Picard groupoids and use this to recover local class field theory for archimedean local fields.

Half of finite abelian groups are unit groups

A group is called realizable if it is the group of units in a ring with identity. The classification of realizable groups is a difficult open problem -- originally posed by László Fuchs -- and is an active area of research. Realizable groups seem rare, but their proportion within a fixed class of groups (cyclic, dihedral, finite abelian, etc.) varies. To quantify this proportion, we introduce the realizable density of a class of finite groups as an analog of natural density for subsets of the natural numbers. The realizable finite cyclic groups and the realizable finite abelian $p$ -groups for $p$ odd have been classified; we prove that their realizable densities are 1/4 and 0, respectively. The realizable finite abelian 2-groups -- and more generally the realizable finite abelian groups -- have not been fully classified, and these special cases appear quite difficult. Nonetheless, we prove that the realizable density of finite abelian 2-groups is 1 and the realizable density of finite abelian groups is 1/2. Our work combines existing classification theorems for realizable groups with tools from analytic number theory.

The Fontaine operator at cusps of modular curves at infinite level

We explicitly calculate Pan's geometric intertwining operator and the Fontaine operator on modular curves at infinite level via $q$ -expansions, using Heuer's theory of cusps at infinite level. We prove that these two operators coincide on such expansions up to an explicit constant. As an application, we combine this result with $q$ -expansion principles to provide a new proof of Pan's theorem that these operators are equal on the locally analytic vectors of completed cohomology of modular curves.

Complete Families of Curves in the Moduli Space of Genus g Curves

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$ . The image of a non-constant morphism from a curve $T$ to $\mathcal{M}_g$ is a curve in $\mathcal{M}_g$ . By work of González Díez and Harvey, for every integer $g \geq 3$ , there exists a complete curve in $\mathcal{M}_g$ . Here we generalize the construction to produce new complete curves in $\mathcal{M}_g$ . We also find a formula for the genus of each curve $T$ using Galois theory for function fields.

Some new results on determinants and permanents

In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\equiv3\pmod 4$ the number $2\det[a_{jk}]_{0\le j,k\le (p-1)/2}$ is congruent to a square modulo $p$ , where $a_{jk}=(\frac{j+k}{p})+(\frac{j^2+k^2}{p})$ with $(\frac{\cdot}{p})$ the Legendre symbol. We also prove that ${\rm per}[j^{k-1}]_{1\leq j,k\leq n-1}\equiv0\pmod n$ for any integer $n>1$ with $n\not\equiv2\pmod 4$ .

A p-adic Labesse-Langlands transfer

We prove a p-adic Labesse-Langlands transfer from the group of units in a definite quaternion algebra to its subgroup of norm one elements. More precisely, given an eigenvariety for the first group, we show that there exists an eigenvariety for the second group and a morphism between them that extends the classical Langlands transfer. In order to find a suitable target eigenvariety for the transfer we formalise a notion of Langlands compatibility of tame levels. Proving the existence of Langlands compatible tame levels is the key to pass from the classical transfer on the level of L-packets to a map between classical points of eigenvarieties, which is then amenable for interpolation to give the p-adic transfer.

preprint2013arXiv

Diophantine approximation and special Liouville numbers

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $ζ_{1},ζ_{2},...,ζ_{k}$ . The approach relies on results on the connection between the set of all $s$ -adic expansions ( $s\geq 2$ ) of $ζ_{1},ζ_{2},...,ζ_{k}$ and their associated approximation constants. As an application, explicit construction of real numbers $ζ_{1},ζ_{2},...,ζ_{k}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.

On uniform approximation to real numbers

Let $n \ge 2$ be an integer and $ξ$ a transcendental real number. We establish several new relations between the values at $ξ$ of the exponents of Diophantine approximation $w_n, w_{n}^{\ast}, \hat{w}_{n}$ , and $\hat{w}_{n}^{\ast}$ . Combining our results with recent estimates by Schmidt and Summerer allows us to refine the inequality $\hat{w}_{n}(ξ) \le 2n-1$ proved by Davenport and Schmidt in 1969.

preprint2015arXiv

Character Sums, Gaussian Hypergeometric Series, and a Family of Hyperelliptic Curves

We study the character sums $ϕ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } ψ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left((x^{m}+a)(x^{n}+b)\right)$ where $ϕ$ is the quadratic character defined over $\mathbb{F}_q$ . These sums are expressed in terms of Gaussian hypergeometric series over $\mathbb{F}_q$ . Then we use these expressions to exhibit the number of $\mathbb{F}_q$ -rational points on families of hyperelliptic curves and their Jacobian varieties.

preprint2015arXiv

Torsion of rational elliptic curves over cubic fields

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.

preprint2014arXiv

Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II

We present an algorithm that computes the Hasse-Witt matrix of given hyperelliptic curve over Q at all primes of good reduction up to a given bound N. It is simpler and faster than the previous algorithm developed by the authors.

preprint2013arXiv

s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ . We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the res

A Novel Proof for Kimberling's Conjecture on Doubly Fractal Sequences

A sequence is a fractal sequence if it contains itself as a proper subsequence. (The self-containment property resembles that of visual fractals) A doubly fractal sequence of integers is defined by operations called upper trimming and lower trimming. C. Kimberling proved that signature sequences are doubly fractal and conjectured the converse. This article gives a procedure for constructing doubly fractal sequences and proves Kimberling's conjecture.

Computing explicit isomorphisms with full matrix algebras over $\mathbb{F}_q(x)$

We propose a polynomial time $f$ -algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$ ) for computing an isomorphism (if there is any) of a finite dimensional $\mathbb{F}_q(x)$ -algebra $A$ given by structure constants with the algebra of $n$ by $n$ matrices with entries from $\mathbb{F}_q(x)$ . The method is based on computing a finite $\mathbb{F}_q$ -subalgebra of $A$ which is the intersection of a maximal $\mathbb{F}_q[x]$ -order and a maximal $R$ -order, where $R$ is the subring of $\mathbb{F}_q(x)$ consisting of fractions of polynomials with denominator having degree not less than that of the numerator.

On the computation of the HNF of a module over the ring of integers of a number field

We present a variation of the modular algorithm for computing the Hermite normal form of an $\mathcal O_K$ -module presented by Cohen, where $\mathcal O_K$ is the ring of integers of a number field $K$ . An approach presented in (Cohen 1996) based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of Cohen to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field $K$ .

Infinite Families of Simple Holomorphic Eta Quotients

We address the problem of constructing a simple holomor- phic eta quotient of a given level N . Such constructions are known for all cubefree N . Here, we provide such constructions for arbitrarily large prime power levels. As a consequence, we obtain an irreducibility

A note on the universality of Hurwitz-Lerch zeta functions

A failed attempt to prove the universality of Lerch zeta function $L(λ,α,s)$ when $λ$ is irrational and $α$ is rational, and for any $λ$ when $α$ is irrational algebraic.

Valuations of $p$ -adic regulators of cyclic cubic fields

We compute the $p$ -adic regulator of cyclic cubic extensions of $\mathbb Q$ with discriminant up to $10^{16}$ for $3<p<100$ , and observe the distribution of the $p$ -adic valuation of the regulators. We find that for almost all primes, the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model, a conjecture on the distribution of the valuations of $p$ -adic regulators of cyclic cubic fields is stated.

On the computation of factorization invariants for affine semigroups

We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

On the $X$ -coordinates of Pell equations which are Tribonacci numbers

For an integer $d\geq 2$ which is not a square, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=\pm 1$ which is a Tribonacci number, with a few exceptions that we completely characterize.

Explicit evaluation of quadratic Euler sums

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.