International Journal of Number Theory

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$,

Let $q$ denote a complex variable with $\left|q\right|<1$. For a positive integer $k$ let

We give an explicit characterization of ergodicity of rational functions on the set of units of the $2$-adic group by means of the coefficients in their nominators and denominators. Although rational functions are not ergodic on the set of p-adic numbers, this was proved by Diao and Silva, we study their ergodicity on the 2-adic unit sphere if they only contain integer coefficients. The first results of this paper provide isometricity tests of polynomials and rational functions on the set of 2-adic units with 2-adic integer coefficients. Then, follow results on ergodicity tests on general isometric functions, then on rational functions on the set of 2-adic units with 2-adic integer coefficients.

We study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes [3]. These representations are symplectic of dimension 4. Following methods used by Dieulefait in [4], we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is “full” for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases.

We study variations of Hodge structures over a Picard modular surface, and compute the weights and types of their degenerations through the cusps of the Baily–Borel compactification. These computations are one of the key inputs which allow Wildeshaus [On the interior motive of certain Shimura varieties: the case of Picard surfaces, Manuscripta Math. 148(3) (2015) 351–377] to construct motives associated with Picard modular forms.

Let $K$ be a number field and let $p$ be an odd rational prime. Let $K_{\infty }$ be a ${\mathbb{Z}}_p$-extension of $K$ and let $L_{\infty }$ be a finite extension of $K_{\infty }$, abelian over $K$. In this paper we extend the classical results, e.g. [16], relating characteristic ideal of the $\chi$-quotient of the projective limit of the ideal class groups to the $\chi$-quotient of the projective limit of units modulo Stark units, in the non-semi-simple case, for some $\overline{{\mathbb{Q}}_p}$-irreductible characters $\chi$ of $\mathrm{\text{Gal}}(L_{\infty }/K_{\infty })$. The proof essentially uses the theory of Euler systems.

We consider certain $p$-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when $p=3$. The proof uses methods from deformation theory and mostly works for any odd prime $p$, but ultimately relies on the existence of a weight $1$ form in an auxiliary family which is available only for $p=3$. We end by giving several non-trivial examples of $3$-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

Starting with a numerically noncritical (at $p$) Hecke eigenclass $f$ in the homology of a congruence subgroup $\mathrm{\Gamma }$ of ${\mathrm{\text{SL}}}_3(\mathbb{Z})$ (where $p$ divides the level of $\mathrm{\Gamma }$) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of $f$ to a Hecke eigenclass $F$ with coefficients in a module of $p$-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection $Z$ to weight space of the eigencurve around the point $z$ corresponding to the system of Hecke eigenvalues of $F$. We do this under the conjecturally mild hypothesis that $Z$ is smooth at $z$.

Let $\varphi$ be a Hecke–Maass cusp form, and $a_{\varphi }(n)$ be its $n$th Fourier coefficient at the cusp infinity. In this paper, we are interested in the estimation on sums

In this paper, we prove that, for any given odd number $k$, the number of integers $n\le x$ such that $k+\varphi (1)+\varphi (2)+\cdots +\varphi (n)$ is a square is $\ll x/{(logx)}^{0.30606}$, where $\varphi (m)$ is the Euler totient function. Beyond this, we pose a conjecture and a problem for further research.

We prove a conjecture of Myerson and Phillips on when an expression involving Dedekind sums is an integer. We also provide a more general result and use this to extend the work of Myerson and Phillips studying whether the points of the graph of the one-variable Dedekind sum that fall on the line are dense.

Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor $loglogn$ for infinitely many dimensions $n$. Here, we prove an effective version of this result, in the sense that we exhibit, for the same set of dimensions, finite families of lattices containing a lattice reaching this bound. Our construction uses codes over cyclotomic fields, lifted to lattices via Construction A. We also prove a similar result for families of symplectic lattices.

We determine the normalizer in SL₂(ℝ) of several families of congruence subgroups of SL₂(ℤ). In addition, we show how these tools can be used to evaluate the groups of automorphisms and the discriminant kernels of a large family of isotropic lattices of signature (2,1).

If $f$ is a polynomial of degree $d$ in ${𝔽}_q[x]$, let $c_k(f)$ be the number of cycles of length $k$ in the directed graph on ${𝔽}_q$ with edges ${\{(v,f(v))\}}_{v\in {𝔽}_q}.$ For random polynomials, the numbers $c_k,1\le k\le b,$ have asymptotic behavior resembling that for the cycle lengths of random functions $f:[q]\to [q].$ However random polynomials differ from random functions in important ways. For example, given the set of cyclic (periodic) points, it is not necessarily true that all permutations of those cyclic points are equally likely to occur as the restriction of $f$. This, and the limitations of Lagrange interpolation, together complicate research on $T(f)=$ the ultimate period of $f$ under compositional iteration. We prove a lower bound for the average value of $logT$: if $d=d(q)\to \infty$, but $d=o(\sqrt{q})$, then the expected value of $logT$ is $𝔼(logT):=\frac{1}{q^d(q-1)}\sum logT(f)>\frac{d}{2}(1+o(1)),$ where the sum is over all $q^d(q-1)$ polynomials of degree $d$ in ${𝔽}_q[x]$. Similar results are proved for rational functions.

A theorem of Mazur gives the set of possible prime degrees for rational isogenies between elliptic curves. In this paper, we are working on a similar problem in the case of abelian surfaces of ${\mathrm{\text{GL}}}_2$-type over $\mathbb{Q}$ with quaternionic multiplication (over $\overline{\mathbb{Q}}$) endowed with a ${\mathrm{\Gamma }}_0(N)$ level structure. We prove the following result: for a fixed indefinite quaternion algebra $B_D$ of discriminant $D$ and a fixed quadratic imaginary field $K=\mathbb{Q}(\sqrt{-d})$, there exists an effective bound $M=O(\text{lcm}{(d,D)}^{22})$ such that for a prime number $N\ge M$, not dividing the conductor of the order ${\mathrm{\text{End}}}_{\mathbb{Q}}(A)$, there do not exist abelian surfaces $A/\mathbb{Q}$ such that ${\mathrm{\text{End}}}_K(A)={\mathrm{\text{End}}}_{\overline{\mathbb{Q}}}(A)$ is a maximal order of $B_D$ and $A$ is endowed with a ${\mathrm{\Gamma }}_0(N)$ level structure.

Let $k,n$ be two positive integers such that $2\le k\le 2^n$ and $S(k,n)$ the numerical semigroup generated by $\{(2^k-1)\cdot 2^{n+i}-1|i\in \mathbb{N}\}$. Then $S(2,n)$ is the Thabit numerical semigroup introduced by J. C. Rosales, M. B. Branco and D. Torrão. In this paper, we give formulas for computing the Frobenius number, the genus and the embedding dimension of $S(k,n)$.

We prove that for a positive integer $k$ the primes in certain kinds of intervals cannot distribute too “uniformly” among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish our results in a much more general situation, in particular for prime ideals in number fields.