We investigate local to global properties for commensurability in Mordell-Weil groups of abelian varieties and tori via reduction maps.

In this paper, we use the contraction mapping principle to obtain mean square asymptotic stability results of a nonlinear stochastic neutral Volterra-Levin equation with Poisson jumps and variable delays. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some previous results due to Burton [5], Becker and Burton [4] and Jin and Luo [10], Ardjouni and Djoudi [1]. Finally, an illustrative example is given.

Inspired by the work of Silverman on the geometry and the arithmetic of monomial maps and also on the translated maps on Abelian varieties, we generalize his results to the case of the translated monomial maps.

An MSTD set is a finite set with more pairwise sums than differences. $(\Upsilon,\Phi)$-ismorphisms are generalizations of Freiman isomorphisms to arbitrary linear forms. These generalized isomorphisms are used to prove that every finite set of real numbers is Freiman isomorphic to a finite set of integers. This implies that there exists no MSTD set $A$ of real numbers with $|A| \leq 7$, and, up to Freiman isomorphism and affine isomorphism, there exists exactly one MSTD set $A$ of real numbers with $|A| = 8$.

Taking the Iwaniec explicit formula as a starting point, we bound the exponent in the error term of the prime geodesic theorem for the modular surface to $2/3$, outside a set of finite logarithmic measure.

We show that the duality relation for the sum of multiple zeta values with fixed weight, depth and $k_1$ is deduced from the derivation relations, which was first conjectured by N. Kawasaki and T. Tanaka.

Let $p\geq 7$ be a prime, $l\geq 0$ be an integer and $k,~m$ be two positive integers, we obtain the following congruences, \[ \sum\limits_{s=lp}^{(l+1)p-1}\binom{kp-1}{s}^m\equiv\begin{cases}\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3},&\text{if}~2\nmid m,\\ \binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}},&\text{if}~2\mid m; \end{cases} \] and \[ \sum\limits_{s=lp}^{(l+1)p-1}(-1)^s\binom{kp-1}{s}^m\equiv\begin{cases}(-1)^l\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3}, &\text{if}~2\mid m,\\ (-1)^l\binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}}, &\text{if}~2\nmid m. \end{cases} \] Let $p$ and $q$ are distinct odd primes and $k$ be a positive integer, we have \[ \binom{kpq-1}{(pq-1)/2}\equiv \binom{kp-1}{(p-1)/2}\binom{kq-1}{(q-1)/2}\pmod {pq}. \]

In this paper, we find the number of representations of the quadratic form $x_1^2+ x_1x_2 + x_2^2 + x_3^2+ x_3x_4 + x_4^2 + \ldots + x_{2k-1}^2 + x_{2k-1}x_{2k} + x_{2k}^2,$ for $k=7,9,11,12,14$ using the theory of modular forms. By comparing our formulas with the formulas obtained by G.A. Lomadze, we obtain the Fourier coefficients of certain newforms of level $3$ and weights $7,9,11$ in terms of certain finite sums involving the solutions of similar quadratic forms of lower variables. In the case of $24$ variables, comparison of these formulas gives rise to a new formula for the Ramanujan tau function.

We use methods of real analysis to continue the Riemann zeta function $\zeta(s)$ to all complex $s$, and to express the values at integers in terms of Bernoulli numbers, using only those infinite series for which we could write down an explicit estimate for the remainder after $N$ terms. This paper is self-contained, apart from appeals to the uniqueness theorems for analytic continuation and for real power series, and, verbis in Latinam translatis, would be accessible to Euler.

Given a fixed Hilbert modular form, we consider a family of linear maps between the spaces of Hilbert cusp forms by using the Rankin-Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Hilbert cusp forms constructed using this method involve special values of certain Dirichlet series of Rankin-Selberg type associated to Hilbert cusp forms.

The solvability over the ring of integers $\mathbb Z$ of some Diophantine equations is connected with the property of integers to form sequences of prime numbers, in particular, with the property of numbers to be twins. The Diophantine description of the sequences of prime numbers is obtained using the deformation method of quadratic matrix equations.