In [Ru15], the author introduced the notion of Nevanlinna constant (denoted by $Nev(D)$) for any effective Cartier divisor $D$ on a normal projective variety $X$, and established a defect relation for Zariski-dense holomorphic mappings $f: {\mathbb C}\rightarrow X$ in terms of $Nev(D)$. In this paper, we prove its counterpart result in Diophantine approximation, according to Vojta's correspondence (or Vojta's dictionary [Voj87]). The results obtained gave the quantitative extension of the earlier results of Corvaja-Zannier [CZ04a,CZ04b], Evertse-Ferretti [EF02, EF08], A. Levin [Lev09], P. Autissier [Aut1], and others.

We present a geometric way of describing the irrationality of a number using the area of a circular sector $A(r)$. We establish a connection between this and the continued fraction expansion of the number, and prove bounds for $A(r)$ as $r\to\infty$ by describing the asymptotic behavior of the ratios of the denominators of the convergents.

In this work we are interested by giving a new $q$-analogue of Bernoulli numbers and polynomials which are symmetric under the interchange $q\leftrightarrow q^{-1}$ and deduce some important relations of them. Also, we deduce a $q$-analogue of the Euler-Maclaurin formulas.

Recently singular overpartitions was defined and studied by G. E. Andrews. He showed that such partitions can be enumerated by $\overline{C}_{\delta,i}(n)$, the number of overpartitions of $n$ such that no part is divisible by $\delta$ and only parts $\equiv\pm i(mod \delta)$ may be overlined. In this paper, we establish several infinite families of congruences $\overline{CO}_{\delta,i}(n)$, the number of singular overpartitions of $n$ into odd parts such that no part is divisible by $\delta$ and only parts $\equiv\pm i\AFODMod{\delta}$ may be overlined. For example, for all $n\geq 0$ and $\alpha\geq0$, $\overline{CO}_{3,1}(4\cdot3^{\alpha+3}n+7\cdot3^{\alpha+2})\equiv 0 \pmod{8}$.

We generalize the function field analog of Gauss' lemma for the application of power residue symbols. We then provide another proof of the general reciprocity law for power residue symbols.

Classical Mean-Value results of Wirsing type are established under weaker than classical constraints.

Désignons par $\lambda_{{\rm sym}^mf}(n)$ le $n$-ème coefficient dans la série de Dirichlet représentant la fonction $L$ de puissances symétriques $L(s, {\rm sym}^mf)$ associée à une forme primitive $f$ de poids $k$ et de niveau $N$. Dans ce papier, on étudie la taille de l'entier le plus petit $n$ tel que $\lambda_{{\rm sym}^mf}(n)<0$ et $(n,N)=1$. En désignant par $n_{{\rm sym}^mf}$ cet entier, on montre que $$ n_{{\rm sym}^3f} \ll (k^{4} N^3)^{6/31} \qquad\text{et}\qquad n_{{\rm sym}^4f} \ll (k^{4} N^4)^{5/36}, $$ où les constantes impliquées sont absolues.

Given a $G$-Galois branched cover of the projective line over a number field $K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a connected fiber such that the\break $G$-Galois field extension induced by specialization ``arithmetically descends'' to $\mathbb{Q}$ (i.e., there exists\break a $G$-Galois field extension of $\mathbb{Q}$ whose compositum with the residue field of the point is equal to the specialization). We prove that the answer is frequently positive (whenever $G$ is regularly realizable over $\mathbb{Q}$) if one first allows a base change to a finite extension of $K$. If one does not allow base change, we prove that the answer is positive when $G$ is cyclic. Furthermore, we provide an explicit example of a Galois branched cover of $\mathbb{P}^1_K$ with no $K$-rational points of arithmetic descent.

We characterize the possible reductions modulo $p$ of the $j$-invariants of supersingular elliptic curves which admit complex multiplication by a (potentially non-maximal) order $\mathcal O$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the collection of possible $j$-invariants as well as some aspects of the distribution depends on which primes divide the discriminant and conductor of the order $\mathcal O$.