Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the “small” solutions of these equations. Under “small” solutions we mean the solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain “totally real” relative Thue equations can be reduced to absolute Thue equations (equations over $\mathbb{Z}$).

As a common generalization of the above results, in our paper we develop a fast algorithm for calculating “small” solutions (say with sizes $\leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the “totally real” case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.

For an extension $K/\mathbb{F}_q(T)$ of the rational function field over a finite field, we introduce the notion of virtually $K$-rational Drinfeld modules as a function field analogue of $\mathbb{Q}$-curves. Our goal in this article is to prove that all virtually $K$-rational Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by $K$-rational points of a quotient curve of the Drinfeld modular curve $Y_0(\mathfrak{n})$ with some square-free level $\mathfrak{n}$. This is an analogue of Elkies' well-known result on $\mathbb{Q}$-curves.

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichletcharacter modulo $\mathfrak{q}$ and $f$ a~primitive holomorphic cusp form or a Hecke-Maass cusp formof level $\mathfrak{q}$and trivial nebentypus. We prove the subconvex bound $$L(1/2,f\otimes \chi)\ll \mathfrak{q}^{1/2-1/12+\varepsilon},$$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifyingtrivial delta methoddeveloped in [1].

We show that there is essentially a unique elliptic curve $E$ defined over a cubic Galois extension $K$ of $\mathbb{Q}$ with a $K$-rational point of order $13$ and such that $E$ is not defined over $\mathbb{Q}$.

W. Kohnen asked whether the values of the classical Dedekind sum are dense in the $p$-adic field $\mathbb{Q}_p$, and K. Girstmair solved this problem. In this paper, we solve the analogous problem for the Dedekind sum in function fields.

Bernoulli-Dunkl and Euler-Dunkl polynomials have been recently introduced as an extension of Bernoulli and Euler polynomials to the Dunkl context. In this article we study the asymptotic behavior of them and prove their convergence to suitable Bessel functions, which are the functions analogous to sine and cosine in the Dunkl context. Finally, we analyze the behavior of the zeros of the Bernoulli-Dunkl and Euler-Dunkl polynomials.

We can improve some previous results related to the distribution of squares in Piatetski–Shapiro sequences of Liu–Shparlinski–Zhang [8].

We consider a multi-variable generalization of the zeta function defined by Arakawa and Kaneko. We establish its analytic continuation to an entire function,whence it follows that values at non-positive integersare expressed as multi-indexed poly-Bernoulli numbers of C-type. We also prove that its values at positive integers are in the space spanned by multiple zeta values.

Jacobi-like forms generalize Jacobi forms, and heat operators on Jacobi forms can be extended to heat operators on Jacobi-like forms. Quasimodular forms correspond to quasimodular polynomials and modular polynomials, and they can be lifted to Jacobi-like forms. We obtain an explicit formula for a differential operator on the space of quasimodular polynomials which is compatible with a heat operators on the space of Jacobi-like forms. We also determine a linear map on the space of modular polynomials that is compatible with this differential operator.