In this paper we prove the meromorphic continuation of the zeta function of Siegel modular threefolds over arbitrary totally real number fields.

In this paper, we prove a relation between the Brauer group and the Tate-Shafarevich group for genus one curves over number fields. This is a generalization of a result of Milne in genus one curves case.

Let $\mathscr{S}_k^+(\mathcal{N},\Phi)$ denote the space generated by Hilbert modular newforms (over a fixed totally real field $K$) of weight $k$, level $\mathcal{N}$ and Hecke character $\Phi$. In this paper we examine the behavior of $\mathscr{S}_k^+(\mathcal{N},\Phi)$ under twists (by a Hecke character). We show how this space may be decomposed into a~direct sum of twists of other spaces of newforms. This sheds light on the behavior of a newform under a~character twist: the exact level of the twist of a newform, when such a~twist is itself a newform, and when a~newform may be realized as the twist of a primitive newform. In certain cases it is shown that the entire space $\mathscr{S}_k^+(\mathcal{N},\Phi)$ can be represented as a direct sum of twists of primitive nebenspaces. This adds perspective to the Jacquet-Langlands correspondence, which characterizes those elements of $\mathscr{S}_k^+(\mathcal{N},\Phi)$ not representable as theta series arising from a quaternion algebra as being precisely those forms which are twists of primitive nebenforms. It follows that in these cases no newforms arise from a quaternion algebra. These results were proven for elliptic modular forms by Hijikata, Pizer and Shemanske by employing the Eichler-Selberg trace formula.

For an odd prime $p$ let $\rho_p$ be the least odd prime ($\ne p$) which is a~quadratic residue mod $p$. Using the theorems of Heegner--Baker--Stark and Siegel--Tatuzawa on the class number $h=h(-p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-p})$ it is shown that $\rho_p<\sqrt p$ unless $p\in \{3, 5, 7, 17, 19, 43, 67, 163\}$, possibly with one further exceptional (large) prime $p=p_u$ (satisfying $p=2^{h+2}-u^2$ with $h>100$ und $5\le u<2^{(h-5)/2}$). The exceptional prime does not exist if the Extended Riemann Hypothesis is true.

For the imaginary quadratic fields with discriminants -120, -132 and -248, the first three, five and two points of the Lagrange and Markov spectra respectively are found.

We introduce the notion of {\em circular words} with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we establish a new result on factors of the Fibonacci word $abaababaabaab\ldots$. Second, we present an expression of the sequence A004146 of [Sloane] in terms of a product of expressions involving roots of unity. Third, we consider the equivalent of $p$-adic numbers that arise by the use of the numeration system defined by the Fibonacci sequence instead of the usual numeration system in base $p$. Among such {\em ${\mathcal F}$-adic numbers}, we give a~characterization of the subset of those which are {\em rational} (that is: a root of an equation of the form $qX=p$, for integral values of $p$ and $q$) by a periodicity property. Eventually, with the help of circular words, we give a complete description of the set of roots of $qX=p$, showing in particular that it contains exactly $q$ ${\mathcal F}$-adic elements.

Let $q, m\geq 2$ be integers with $(m,q-1)=1$. Denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Further let $p(x)\in \mathbb{Z}[x]$ be a polynomial of degree $h\geq 3$ with $p(\mathbb{N})\subset \mathbb{N}$. We show that there exist $C=C(q,m,p)>0$ and $N_0=N_0(q,m,p)\geq 1$, such that for all $g\in\mathbb{Z}$ and all $N\geq N_0$, $$\#\{0\leq n< N: s_q(p(n))\equiv g \bmod m\}\geq C N^{4/(3h+1)}.$$ This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is $C N^{2/h!}$.

In this paper, we study representations of real numbers in the positional numeration system with negative base, as introduced by Ito and Sadahiro. We focus on the set $\mathbb{Z}_{-\beta}$ of numbers whose representation uses only non-negative powers of $-\beta$, the so-called $(-\beta)$-integers. We describe the distances between consecutive elements of $\mathbb{Z}_{-\beta}$. In case that this set is non-trivial we associate to $\beta$ an infinite word $\boldsymbol{v}_{-\beta}$ over an (in general infinite) alphabet. The self-similarity of $\mathbb{Z}_{-\beta}$, i.e., the property $-\beta \mathbb{Z}_{-\beta}\subset \mathbb{Z}_{-\beta}$, allows us to find a~morphism under which $\boldsymbol{v}_{-\beta}$ is invariant. On the example of two cubic irrational bases $\beta$ we demonstrate the difference between Rauzy fractals generated by $(-\beta)$-integers and by $\beta$-integers.

For given positive integers $a,b,q$ we investigate the density of solutions $(x,y)\in \mathbb{Z}^2$ to congruences $ax+by^2\equiv 0 \bmod{q}$.