We complete, in this paper, the characterization of all known even perfect polynomials over the prime field $\mathbb{F}_2$. In particular, we prove that the last two of the eleven known ``sporadic'' perfect polynomials over $\mathbb{F}_2$ are the unique of them of the form $x^a(x+1)^b M^{2h} \sigma(M^{2h})$, where $M$ is a Mersenne prime and $a,b, h \in \mathbb{N}^*$.

In this article we study the following problem. Let $k$ be an infinite field and let $f \in k[x]$. Consider the evaluation map $f_k: k \to k$. Assume that $f_k$ is not surjective. Is $k \smallsetminus f_k(k)$ infinite? We give a positive answer to this question when $k$ is a perfect ample field. In fact, we prove that $|k \smallsetminus f_k(k)|=|k|$. This conclusion follows from a similar statement about finite morphisms between normal projective curves over perfect ample fields.

We prove the mean square formula of the Euler--Zagier type double zeta-function $\zeta_{2}(s_{1},s_{2})$ and provide an improvement on the $\Omega$ results of Kiuchi, Tanigawa, and Zhai. We also calculate the double integral $\int_{2}^{N}\int_{2}^{T}|\zeta_{2}(s_{1},s_{2})|^{2}dt_{1} dt_{2}$ under certain conditions.

Let $f$ be a cusp form of integral weight $k \geq 4$ for $\Gamma_{0}(N)$ with nebentypus $\psi$. Generalising work of Kohnen we construct a kernel function for the $L$-function $L(f,\chi,s)$ of $f$ twisted by a primitive Dirichlet character $\chi$ and use it to show that the average $\sum_{f \in S_{k}(N,\psi)}\frac{L(f,\chi,s)}{\langle f,f\rangle}\overline{a_{f}(1)}$ over an orthogonal basis of $S_{k}(N,\psi)$ does not vanish on certain rectangles inside the critical strip if the weight $k$ or the level $N$ is big enough. As another application of the kernel function we prove an averaged version of Waldspurger's Theorem.

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than $\mathbb{Z}$. We then study the growth of the $p^\infty$-Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers $k\subset K\subset F$ in which $[F:K]$ is not a $p$-power extension.

In 1952 Paul Erdős obtained upper and lower bounds of the same order of magnitude for the number $N(x)$ of divisors of an irreducible polynomial $f(n)$ with integer coefficients for $n$ up to $x$; an asymptotic formula for $N(x)$ when $f$ has degree at least $3$ has not yet been established. However progress has been made in the corresponding problem when the divisors of $f(n)$ are restricted in some way and $f$ is not necessarily irreducible. In this paper we consider a~polynomial $f$ $ $with integer coefficients that may not be irreducible or squarefree. Our aim is to obtain an asymptotic formula for the number of exact divisors up to $y$ of $f(p)$ for $p$ a prime less than $x$ with $y$ as large as possible in terms of $x$. We utilize the result that Vaughan established for his elementary proof of the Bombieri-Vinogradov Theorem.

We compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order two, and of the stack of elliptic curves equipped with a cyclic subgroup of order three, over any base scheme on which $6$ is invertible. This generalizes a result of Fulton-Olsson, who computed the Picard group of the stack of elliptic curves (with no level structure) over a~wide variety of base schemes.

In this paper, we will study some properties of the extended Eulerian numbers $H(n,\lambda)$, with a parameter $\lambda$. In fact, for any integer $n$, we investigate the asymptotic behavior, find lower and upper bounds for $H(n,\lambda)$. As application, for a champion number $N$, we obtain asymptotic formulas, lower and upper bounds of the arithmetic functions $\omega(N)$ and $\Omega(N)$.

Let $k \ge 6$. Using the recent result of Bourgain, Demeter, and Guth \cite{1586:bdg} on the Vinogradov mean value, we obtain new bounds for small fractional parts of polynomials $\alpha_kn^k + \cdots + \alpha_1n$ and additive forms $\beta_1n_1^k + \cdots + \beta_sn_s^k$. Our results improve earlier theorems of Danicic (1957), Cook (1972), Baker (1982, 2000), Vaughan and Wooley (2000), and Wooley (2013).