Let $\theta(q)=1+2\sum_{\nu=1}^{\infty} q^{\nu^2}$ denote the Thetanullwert of the Jacobi Zeta function \[\theta(z|\tau) =\sum_{\nu=-\infty}^{\infty} e^{\pi i\nu^2\tau + 2\pi i\nu z}.\] For algebraic numbers $q$ with $0<|q|<1$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers $\theta(q^n)$ and $\theta(q)$ for $n=2,3,\dots,12$ and furthermore for all $n\geq 16$ which are powers of two. An application for $n=5$ proves the transcendence of the number \[\sum_{j=1}^{\infty} {(-1)}^j \Big( \frac{j}{5}\Big) \frac{jq^j}{1-q^j}.\] Similar results are obtained for numbers related to modular equations of degree 3, 5, and 7.

H. Rademacher and E. Grosswald raised the following questions: \begin{enumerate} \item Is $\{ (a/c, d(a,c))\ |\ a/c\in\mathbb{Q}^*\}$ dense in $\mathbb{R}^2$? \item Is $\{ d(a,c)\ |\ a/c\in\mathbb{Q}^*\}$ dense in $\mathbb{R}$? \end{enumerate} D. Hickerson answered them affirmatively, and H. Ito obtained a result similar to Hickerson's for the elliptic Dedekind sums defined by R. Sczech. We consider the values of the Dedekind sum attached to a given $A$-lattice in rational function fields. The objective of this paper is to establish a result similar to those of Hickerson and Ito.

We prove that for any given positive integer $\ell$ there are infinitely many imaginary quadratic fields with 2-class group of type $(2,2^\ell)$, and provide a lower bound for the number of such groups with bounded discriminant for $\ell\ge2$. This work is based on a related result for cyclic 2-class groups by Dominguez, Miller and Wong, and our proof proceeds similarly. Our proof requires introducing congruence conditions into Perelli's result on Goldbach numbers represented by polynomials, which we establitish in some generality.

In this paper we prove Tate conjecture for abelian surfaces of the type $\operatorname{Res}_{K/F}E$ where $E$ is an elliptic curve defined over a totally real or CM number field $K$, and $F$ is a subfield of $K$ such that $[K:F]=2$.

We consider the linear vector space formed by the elements of the finite field $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Then the elements $x$ of $\mathbb{F}_q$ have a unique representation in the form $x=\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$; the coefficients $c_j$ will be called digits. Let $\mathbb{D}$ be a subset of $\mathbb{F}_p$ with $2\le |\mathbb{D}|<p$. We consider elements $x$ of $\mathbb{F}_q$ such that for their every digit $c_j$ we have $c_j\in\mathbb{D}$; then we say that the elements of $\mathbb{F}_p\setminus\mathbb{D}$ are ``missing digits''. We will show that if $\mathbb{D}$ is a large enough subset of $\mathbb{F}_p$, then there are squares with missing digits in $\mathbb{F}_q$; if the degree of the polynomial $f(x)\in\mathbb{F}_q[X]$ is at least $2$ then it assumes values with missing digits; there are generators $g$ in $\mathbb{F}_q$ such that $f(g)$ is of missing digits.

We construct an operator on modular forms by modifying the derivative operator using certain theta series. To show that the image of a~modular form under this operator is a~modular form we use Jacobi-like forms determined by theta series as well as quasimodular forms obtained from such Jacobi-like forms.

We investigate linear relations in Mordell-Weil groups of abelian varieties over finitely generated fields over $\mathbb{Q}$. Based on important and classical results for abelian varieties over these fields and on lifts of abelian varieties to suitable abelian schemes, we prove theorems concerning the reduction maps on torsion and non-torsion elements in Mordell-Weil groups of these varieties. These theorems and the arithmetic of abelian schemes and their endomorphism algebras are our key tools in the solutions of linear relation problems we work with in the last chapter of this paper.

We study special values of Artin $L$-functions for dihedral extensions at negative integers. We give a relation between these values and orders of the $\chi$-parts of certain étale cohomology groups.

In this paper, we give criteria for infinitely many sign changes of the coefficients of any Dirichlet series if the coefficients are real numbers. We also provide examples where our criteria are applicable.

In this short note we show that the uniform $abc$-conjecture puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of Vojta's height inequality formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.

We make a systematic study of van der Corput's $B$-process for multiple exponential sums. We study directly the important case where the determinant of the Hessian $H_{f}(\mathbf{x})$ of the phase $f$ may be abnormally small. This requires a work on multidimensional stationary phase integrals uniform in $\delta$, the lower bound for $||\det H_{f}(\mathbf{x})||$. In passing, we obtain an independent result on the asymptotic behaviour of the stationary phase integral when the critical point of the phase is also a singular point of the boundary of the domain of integration. The whole paper is self-contained.