Mathematics - Number Theory Publications (50)


Mathematics - Number Theory Publications

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. Read More

The elliptic modular j function enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. Read More

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists over its minimal field of definition. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. Read More

Let $F$ and $G$ be linear recurrences over a number field $\mathbb{K}$, and let $\mathfrak{R}$ be a finitely generated subring of $\mathbb{K}$. Furthermore, let $\mathcal{N}$ be the set of positive integers $n$ such that $G(n) \neq 0$ and $F(n) / G(n) \in \mathfrak{R}$. Under mild hypothesis, Corvaja and Zannier proved that $\mathcal{N}$ has zero asymptotic density. Read More

In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications to problems in Diophantine approximation. These include a general transference of Khintchine-Groshev type theorems into Hausdorff measure statements. Read More

In the present paper, we introduce some new families of elliptic curves with positive rank arrising from Pythagorean triples. Read More

We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $\mathbb{S} \to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $\pi_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra. Read More

The Kuznetsov and Petersson trace formulae for $GL(2)$ forms may collectively be derived from Poincar\'e series in the space of Maass forms with weight. Having already developed the spherical spectral Kuznetsov formula for $GL(3)$, the goal of this series of papers is to derive the spectral Kuznetsov formulae for non-spherical Maass forms and use them to produce the corresponding Weyl laws; this appears to be the first proof of the existence of such forms not coming from the symmetric-square construction. Aside from general interest in new types of automorphic forms, this is a necessary step in the development of a theory of exponential sums on $GL(3)$. Read More

We construct two families of harmonic Maass Hecke eigenforms. This construction answers a question of Mazur about the existence of an "eigencurve-type" object in the world of harmonic Maass forms. Using these families, we construct $p$-adic harmonic Maass forms in the sense of Serre. Read More

We describe a decomposition algorithm for elliptic multiple zeta values, which amounts to the construction of an injective map $\psi$ from the algebra of elliptic multiple zeta values to a space of iterated Eisenstein integrals. We give many examples of this decomposition, and conclude with a short discussion about the image of $\psi$. It turns out that the failure of surjectivity of $\psi$ is in some sense governed by period polynomials of modular forms. Read More

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312. Read More

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Read More

We study the algebra $\mathcal{E}$ of elliptic multiple zeta values, which is an elliptic analog of the algebra of multiple zeta values. We identify a set of generators of $\mathcal{E}$, which satisfy a double shuffle type family of algebraic relations, similar to the double-shuffle relations for multiple zeta values. We prove that the elliptic double shuffle relations give all algebraic relations among elliptic multiple zeta values, if (a) the classical double shuffle relations give all algebraic relations among multiple zeta values and if (b) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure. Read More

Consider the power pseudorandom-number generator in a finite field ${\mathbb F}_q$. That is, for some integer $e\ge2$, one considers the sequence $u,u^e,u^{e^2},\dots$ in ${\mathbb F}_q$ for a given seed $u\in {\mathbb F}_q^\times$. This sequence is eventually periodic. Read More

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Read More

We explicitly determine the values of reduced cyclotomic periods of order $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. These evaluations are applied to obtain explicit factorizations of the corresponding reduced period polynomials. As another application, the weight distributions of certain irreducible cyclic codes are described. Read More

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Read More

We estimate the maximum-order complexity of a binary sequence in terms of its correlation measures. Roughly speaking, we show that any sequence with small correlation measure up to a sufficiently large order $k$ cannot have very small maximum-order complexity. Read More

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin $L$-series at $s=r$ to the compact support cohomology of the \'etale $p$-adic sheaf $\mathbb Z_p(r)$. Read More

Schmidt's game is generally used to to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the "middle $\epsilon$" Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimension of the set of $\epsilon$-badly approximable numbers on the Cantor set? We show that a variant of Schmidt's game known as the $potential$ $game$ is capable of providing better bounds on the answers to these questions than the classical Schmidt's game. Read More

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk. Read More

We show the middle Nth cantor set contains arithmetic progressions of length at least proportional to N/log_2(N). Read More

In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the $j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax-Schanuel conjecture. In this article, we seek to generalize the ideas of Habegger and Pila to show that, under a number of arithmetic hypotheses, the hyperbolic Ax-Schanuel conjecture can faciltate an extension of the Pila-Zannier strategy to the Zilber-Pink conjecture for Shimura varieties. Read More

In this paper, we estimate the shifted convolution sum \[\sum_{n\geqslant1}\lambda_1(1,n)\lambda_2(n+h)V\Big(\frac{n}{X}\Big),\] where $V$ is a smooth function with support in $[1,2]$, $1\leqslant|h|\leqslant X$, $\lambda_1(1,n)$ and $\lambda_2(n)$ are the $n$-th Fourier coefficients of $SL(3,\mathbf{Z})$ and $SL(2,\mathbf{Z})$ Hecke-Maass cusp forms, respectively. We prove an upper bound $O(X^{\frac{21}{22}+\varepsilon})$, updating a recent result of Munshi. Read More

Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodsky's h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusie's de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. Read More

Let $L$ be a degree-$2$ $L$-function associated to a Maass cusp form. We explore an algorithm that evaluates $t$ values of $L$ on the critical line in time $O(t^{1+\varepsilon})$. We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution. Read More

The aim of this article is to investigate the pairs of positive-definite real ternary quadratic forms that have perfectly identical representations over Z. The result of an exhaustive search for such pairs of integral quadratic forms is presented. It shows that there is some limitation on the existence of such pairs, regardless of whether the coefficient field is Q or R. Read More

We improve on previous upper bounds for the $q$th norm of the partial sums of the Riemann zeta function on the half line when $0Read More

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solutions of certain functional equations. Read More

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)^r}{p^{\alpha}}\right)\right). Read More

We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero $\rho$ (trivial or nontrivial) and each zeta fixed point $\psi$ there is a nearly logarithmic spiral $s_{\rho, \psi}$ with center $\psi$ containing $\rho$. (2) $s_{\rho, \psi}$ interpolates a subset $B_{\rho, \psi}$ of the backward zeta orbit of $\rho$ comprising a set of zeros of all iterates of zeta. (3) If zeta is viewed as a function on sets, $\zeta(B_{\rho, \psi}) = B_{\rho, \psi} \cup \{0 \}$. Read More

A lattice in the Euclidean space is standard if it has a basis consisting vectors whose norms equal to the length in its successive minima. In this paper, it is shown that with the $L^2$ norm all lattices of dimension $n$ are standard if and only if $n\leqslant 4$. It is also proved that with an arbitrary norm, every lattice of dimensions 1 and 2 is standard. Read More

We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are "reflective" versions of the complex continued fractions of A. L. Read More

In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as (reciprocal) logarithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of the second kind. In addition, two interesting series with rational terms are given for Euler's constant and the constant ln(2*pi), and yet another generalization of Euler's constant is proposed and various formulas for the calculation of these constants are obtained. Read More

We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on the rational dynamics. Read More

We establish a refinement of Marstrand's projection theorem for Hausdorff dimension functions finer than the usual power functions, including an analogue of Marstrand's Theorem for logarithmic Hausdorff dimension. Read More

Let $f(z)=\sum_{n=1}^{\infty}a(n) e^{2\pi i nz}$ be a normalized Hecke eigenform in $S_{2k}^{\text{new}}(\Gamma_0(N))$ with integer Fourier coefficients. We prove that there exists a constant $C(f)>0$ such that any integer is a sum of at most $C(f)$ coefficients $a(n) $. It holds $C(f)\ll_{\varepsilon,k}N^{\frac{6k-3}{16}+\varepsilon}$. Read More

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points of E. An explicit description of E(A) is given, and we prove that for K of degree n, almost all elliptic curves over K have an adelic point group topologically isomorphic to a universal group depending on n. We also show that there exist infinitely many elliptic curves over K having a different adelic point group. Read More

We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics. Moreover, we give a complete description of the moduli of these surfaces. Read More

In this note, we investigate properties of the ratio $D(n)/n$, which we will call the deficiency index. We will discuss some concepts recast in the language of the deficiency index, based on similar considerations in terms of the abundancy index. Read More

We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for $\Gamma_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. Read More

A complex projective manifold is rationally connected if every finite subset is connected by a rational curve. For a "rationally simply connected" manifold, the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field with vanishing "elementary obstruction" has a rational point if it deforms to a rationally simply connected variety in characteristic 0. Read More

We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. Read More

An irreducible smooth projective curve over $\mathbb{F}\_q$ is called \textit{pointless} if it has no $\mathbb{F}\_q$-rational points. In this paper we study the lower existence bound on the genus of such a curve over a fixed finite field $\mathbb{F}\_q$. Using some explicit constructions of hyperelliptic curves, we establish two new bounds that depend linearly on the number $q$. Read More

Using the rank of the Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve $E$ over $\mathbb{Q}$, we give a lower bound of the class number of the number field $\mathbb{Q}(E[p^n])$ generated by $p^n$-division points of $E$ when the curve $E$ does not possess a $p$-adic point of order $p$: $E(\mathbb{Q}_p)[p] =0$. Read More

We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $\mathbb{Q}$. Read More

In a former paper the authors introduced two new systematic authentication codes based on the Gray map over a Galois ring. In this paper, it is proved the one-to-one onto correspondence between keys and encoding maps for the second introduced authentication code. Read More

Let $M_k^\sharp(N)$ be the space of weight $k$, level $N$ weakly holomorphic modular forms with poles only at the cusp at $\infty$. We explicitly construct a canonical basis for $M_k^\sharp(N)$ for $N\in\{8,9,16,25\}$, and show that many of the Fourier coefficients of the basis elements in $M_0^\sharp(N)$ are divisible by high powers of the prime dividing the level $N$. Additionally, we show that these basis elements satisfy a Zagier duality property, and extend Griffin's results on congruences in level 1 to levels 2, 3, 4, 5, 7, 8, 9, 16, and 25. Read More

We attach buildings to modular lattices and use them to develop a metric approach to Harder-Narasimhan filtrations. Switching back to a categorical framework, we establish an abstract numerical criterion for the compatibility of these filtrations with tensor products. We finally verify our criterion in three cases, one of which is new. Read More

In this paper we first establish new explicit estimates for Chebyshev's $\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime counting function $\pi(x)$, which improve the currently best ones. Furthermore, we use the obtained estimates for the prime counting function to give two new results concerning the existence of prime numbers in short intervals. Read More