We construct explicit examples of genus two fibrations with no sections on rational surfaces by the double covering method. For the proof of non-existence of sections, we use the theory of the virtual Mordell-Weil groups.

We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a proof for its convergence. The result implies that there are cases of convergence of finite difference schemes applied to ill-posed Cauchy problems.

Let $F$ be a non-archimedean local field of arbitrary characteristic. In this paper, we announce an explicit formula of the unramified Shintani functions for $(\mathbf{GSp}_{4}(F),(\mathbf{GL}_{2} \times_{\mathbf{GL}_{1}} \mathbf{GL}_{2})(F))$. As an application, we compute a local zeta integral, which represents the spin $L$-factor of $\mathbf{GSp}_{4}$.

In 1906, Maillet proved that given a non-constant rational function $f$, with rational coefficients, if $\xi$ is a Liouville number, then so is $f(\xi)$. Motivated by this fact, in 1984, Mahler raised the question about the existence of transcendental entire functions with this property. In this work, we define an uncountable subset of Liouville numbers for which there exists a transcendental entire function taking this set into the set of the Liouville numbers.