International Journal of Number Theory

We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus $n$ to all the rational boundaries of lower degrees. This global operator reduces to Siegel's $\mathrm{\Phi }$ operator when considering only the maximal standard cusps of degree $n-1$. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus $n>1$ and to bridge this new notion with the classical one found in the literature.

Dedekind used the classical Dedekind sum $D(a,c)$ to describe the transformation of $log\eta (z)$ under the substitution $z^{\prime }=(az+b)/(cz+d),(\begin{array}{cc}a\hfill & b\hfill \\ c\hfill & d\hfill \end{array})\in {\mathrm{\text{SL}}}_2(\mathbb{Z})$. In this paper, we use the Dedekind sum $s(a,c)$ in function fields to describe the transformation of a certain series under the substitution $z^{\prime }=(az+b)/(cz+d),(\begin{array}{cc}a\hfill & b\hfill \\ c\hfill & d\hfill \end{array})\in {\mathrm{\text{GL}}}_2({𝔽}_q[T])$.

Let $a>1$ be an integer. Denote by $l_a(n)$ the multiplicative order of $a$ modulo integer $n\ge 1$. We prove that there is a positive constant $\delta$ such that if $x^{1-\delta }=o(y)$, then

Mordell in 1958 gave a new proof of the three squares theorem. We generalize those techniques to characterize the integers represented by the remaining six “Ramanujan–Dickson ternaries” as well as three other ternary forms.

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms with sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight $k\ge 0$, which is related to the weight of Borcherds lifts when $k=0$. In particular, we see that such divisibility of the weight of Borcherds lifts only exists for $\mathbb{Q}(\sqrt{5})$. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, we obtain divisibility results in an “orthogonal” direction on reduced modular forms.

The theorems of Gauss and Jacobi that give modulo $p$ evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo $p^2$. In this paper, we further extend these results to congruences modulo $p^3$. In the process, we prove congruences to arbitrarily high powers of $p$ for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's $p$-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

Let $\mathrm{\Gamma }$ be a finitely generated subgroup of ${\mathbb{Q}}^{\ast }$. Let $p$ be a prime number for which the reduction group ${\mathrm{\Gamma }}_p$ is a well-defined subgroup of the multiplicative group ${(\mathbb{Z}/p\mathbb{Z})}^{\ast }$, and denote the order of ${\mathrm{\Gamma }}_p$ by $\left|{\mathrm{\Gamma }}_p\right|$. Assuming the Generalized Riemann Hypothesis, we study the average of $\left|{\mathrm{\Gamma }}_p\right|$ and the average of the powers of $\left|{\mathrm{\Gamma }}_p\right|$ as $p$ ranges over prime numbers. The problem was considered in the case of rank 1 by Pomerance and Kurlberg. In the case when $\mathrm{\Gamma }$ contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.

In this note, we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue–Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.

Let $b,n\in \mathbb{Z}$, $n\ge 1$, and ${{𝒟}}_1,\dots ,{{𝒟}}_{\tau (n)}$ be all positive divisors of $n$. For $1\le l\le \tau (n)$, define ${{𝒞}}_l:=\{1\le x\le n:(x,n)={{𝒟}}_l\}$. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence $x_1+\cdots +x_k\equiv b(modn)$, with ${\kappa }_l=|\{x_1,\dots ,x_k\}\cap {{𝒞}}_l|$, $1\le l\le \tau (n)$, is

We study the character sums

In this paper, we study finite forms of Bailey's transformation formulae and therefrom, we deduce the Bailey's transformation formulae. Further, we derive finite forms of the complete mock theta functions of second order. Also we give a simple proof of one of Bailey's transformation formulae and present some applications.

Let $K$ be a number field, $𝔮$ be an integral ideal, and $\mathrm{\text{Cl}}(𝔮)$ be the associated narrow ray class group. Suppose $\mathrm{\text{Cl}}(𝔮)$ possesses a real exceptional character $\psi$, possibly principal, with a Siegel zero $\beta$. For ${𝒞}\in \mathrm{\text{Cl}}(𝔮)$ satisfying $\psi$ $({𝒞})=1$, we establish an effective $K$-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal $𝔭\in {𝒞}$. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.

Let $d(n)$ be the number of divisors of $n$, let

Let $K$ be a complete discrete valuation field. Let ${{𝒪}}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\ge 1$. Let $G/K$ be a semi-abelian variety. Let ${𝒢}/{{𝒪}}_K$ be its Néron model. The special fiber ${{𝒢}}_k/k$ is an extension of the identity component ${{𝒢}}_k^0/k$ by the group of components $\mathrm{\Phi }(G)$. We say that $G/K$ has split reduction if this extension is split. Whereas $G/K$ has always split reduction if $p=1$ we prove that it is no longer the case if $p>1$ even if $G/K$ is tamely ramified. If $J/K$ is the Jacobian variety of a smooth proper and geometrically connected curve $C/K$ of genus $g$, we prove that for any tamely ramified extension $M/K$ of degree greater than a constant, depending on $g$ only, $J_M/M$ has split reduction. This answers some questions of Liu and Lorenzini.

Let $K$ be a number field, $f\in K[x]$ a quadratic polynomial, and $n\in \{1,2,3\}$. We show that if $f$ has a point of period $n$ in every non-archimedean completion of $K$, then $f$ has a point of period $n$ in $K$. For $n\in \{4,5\}$ we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over $K$ for which this local–global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn–Poonen–Schaefer in the case $K=Q$. More precisely, we show that for every quadratic polynomial $f\in Q[x]$ there exist infinitely many primes $p$ such that $f$ does not have a point of period four in the $p$-adic field $Q_p$. Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period five.

Let $\chi$ be a primitive Dirichlet character of conductor $q$ and let us denote by $L(z,\chi )$ the associated $L$-series. In this paper, we provide an explicit upper bound for $|L(1,\chi )|$ when $\chi$ is a primitive even Dirichlet character with $\chi (2)=1$.

Let $K=\mathbb{Q}(\theta )$ be an algebraic number field with $\theta$ in the ring $A_K$ of algebraic integers of $K$ and $F(x)$ be the minimal polynomial of $\theta$ over the field $\mathbb{Q}$ of rational numbers. In 1977, Uchida proved that $A_K=\mathbb{Z}[\theta ]$ if and only if $F(x)$ does not belong to ${𝔐}^2$ for any maximal ideal $𝔐$ of the polynomial ring $\mathbb{Z}[x]$ (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of $F(x)$ for $A_K$ to equal $\mathbb{Z}[\theta ]$ when $F(x)$ is a trinomial of the type $x^n+ax+b$. In the particular case when $a=-1$, it is deduced that $\{1,\theta ,\dots ,{\theta }^{n-1}\}$ is an integral basis of $K$ if and only if either (i) $p\nmid b$ and $p^2\nmid (b^{n-1}{n}^n-{(n-1)}^{n-1})$ or (ii) $p$ divides $b$ and $p^2\nmid b$.

Let $m$ be a positive integer, and define