We prove Kronecker's first limit formula for cofinite discrete subgroups of SL$(2,\mathbbC)$, also called Kleinian groups, generalizing a method of Goldstein over SL$(2,\mathbb R)$. The proof uses the Fourier expansion of Eisenstein series, leading to an analogue of the logarithm of the Dedekind eta function, and whose transformation law produces a function analogous to the one related to Dedekind sums. We derive several basic properties of this new function.

In this paper, series of modulus of differences of general Fourier coefficients are studied. This way we define the bounded variation of general Fourier coefficients of the function from some functional class. Throughout the paper we show that the sequence of general Fourier coefficients even of the function $g(x)=1$ is not of bounded variation. We find a condition on the functions of an orthonormal system (ONS), under which the sequence of Fourier coefficients of functions is of bounded variation, when the function is from the bounded variation class. We also study the behaviour of Fourier coefficients with respect to the classical ONS (trigonometric [1, Ch.1, §2], Haar [1, Ch. 1], Walsh [2] systems). In addition, we investigate the subsequences of ONS.

Suppose that the rational right triangle triple is $(T_1,T_2,T_3)$, their areas are $A_i~(i=1,2,3)$, and perimeters are $P_i~(i=1,2,3)$. By the theory of elliptic curves, we investigate the solvability of the following Diophantine system \[A_1+\alpha A_2=\beta A_3,\qquad P_1+\alpha P_2=\beta P_3,\]where $\alpha$ and $\beta$ are rational numbers. When $(\alpha,\beta)=(-2,-1)$ or $(\alpha,\beta)=(1,1)$, we show that there are infinitely many rational right triangle triples with the same perimeter and the areas in arithmetical progression or with the areas and perimeters satisfying the linear recurrence equation of Lucas sequence respectively. Moreover, we prove that there is no rational right triangle triple whose areas, perimeters and radii of the inscribed circles satisfy the linear recurrence equation of Lucas sequence respectively.

We consider the product sequences of the sequences $(\Psi _{n}(\mathbf P))$, $(\Phi _{n}(\mathbf P))$, and $(\overline{\Omega}_{n}(\mathbf P))$ ($n\in \mathbb N$) of values of the translated division polynomials of an elliptic curve $E/K $ evaluated at a point $\mathbf P\in $ $E(K)^{2}$. We prove that thesesequences are purely periodic when $K$ is a finite field. Then we use $p$-adic properties of these sequences to obtain $p$-adic convergence ofproduct of the Somos $4$ and Somos $5$ sequences.

We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions $_{n+1}F_n$, $n\ge 1$, over finite fields with $q$ elements where $q$ is an odd prime. This enables us to find an estimate for the value $_6F_5(1)$. In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value $_3F_2(-1)$. These formulas also allow us to express the product of certain $_2F_1$ and $_{n+1}F_n$ functions in terms of finite field Appell series which generalizes current formulas for products of $_2F_1$ functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.

Let $f(x)\in \mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $\mathbb{Q}$. We say $f(x)$ is \emph{monogenic} if $\Theta=\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers $\mathbb{Z}_K$ of $K=\mathbb{Q}(\theta)$, where $f(\theta)=0$. If $\Theta$ is not a basis for $\mathbb{Z}_K$, we say that $f(x)$ is \emph{non-monogenic}.Let $k\ge 1$ be an integer, and let $(U_n)$be the sequence defined by \[U_0=U_1=0,\qquad U_2=1 \qquad \text{and}\qquad U_n=kU_{n-1}+(k+3)U_{n-2}+U_{n-3} \qquad \text{for $n\ge 3$}.\] It is well known that $(U_n)$ is periodic modulo any integer $m\ge 2$, and we let $\pi(m)$ denote the length of this period. We define a \emph{$k$-Shanks prime} to be a prime $p$ such that $\pi(p^2)=\pi(p)$. Let $\mathcal{S}_k(x)=x^{3}-kx^{2}-(k+3)x-1$ and $\mathcal{D}=(k^2+3k+9)/\gcd(3,k)^2$. Suppose that $k\not \equiv 3 \pmod{9}$ and that $\mathcal{D}$ is squarefree. In this article, we prove that $p$ is a $k$-Shanks prime if and only if $\mathcal{S}_k(x^p)$ is non-monogenic, for any prime $p$ such that $\mathcal{S}_k(x)$ is irreducible in $\mathbb{F}_p[x]$. Furthermore, we show that $\mathcal{S}_k(x^p)$ is monogenic for any prime divisor $p$ of $\mathcal{D}$. These results extend previous work of the author on $k$-Wall-Sun-Sun primes.

In this paper we prove the Hasse principle for polynomials with coefficients in Mordell-Weil type groups over number fields. Examples of such groups are (1) the groups of $S$-units in a number field, (2) abelian varieties with trivial ring of endomorphisms, and (3) odd algebraic $K$-theory groups.

The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of $\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$ where $n$ is an integer. We know that, in general, we can realize these groups as the Galois group of a given number field, using the torsion points on an elliptic curve. Specifically, a theorem of Reverter and Vila gives, for each prime $n$, a polynomial, depending on an elliptic curve, whose Galois group is $\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$. In this article, we generalize this theorem in several directions, in particular for $n$ not necessarily prime. We also determine a minimum for the valuations of the coefficients of the polynomials arising in our construction, depending only on $n$.