We establish new results on complex and $p$-adic linear independence on a class of semiabelian varieties. As applications, we obtain transcendence results concerning complex and $p$-adic Weierstrass sigma functions associated with elliptic curves.

We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre p-adique de variétés de groupe’, Invent. Math. 40(2) (1977), 171–193].

We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

The proof of Theorem 3.2 in [1] (P. Tzermias, On the p-adic binomial series and a formal analogue of Hilbert's Theorem 90, Glasgow Math. J.47 (2005), 319–326) contains two opaque claims. The necessary clarifications are provided here.

Let p be a prime number, Qp the field of p-adic numbers, K a finite field extension of Qp, K a fixed algebraic closure of K and Cp the completion of K with respect to the p-adic valuation. We introduce and investigate an equivalence relation on Cp, defined in terms of field extensions and metric properties of Galois orbits over K.

it is conjectured that a rational map whose coefficients are algebraic over $\mathbb q_p$ has no wandering components of the fatou set. benedetto has shown that any counterexample to this conjecture must have a wild recurrent critical point. we provide the first examples of rational maps whose coefficients are algebraic over $\mathbb q_p$ and that have a (wild) recurrent critical point. in fact, it is shown that there is such a rational map in every one-parameter family of rational maps that is defined over a finite extension of $\mathbb q_p$ and that has a misiurewicz bifurcation.

Let $p$ be a prime number, $\Q_p$ the field of $p$-adic numbers, $K$ a finite field extension of $\Q_p$, $\skew4\bar K$ a fixed algebraic closure of $K$, and $\C_p$ the completion of $\skew4\bar K$ with respect to the $p$-adic valuation. We discuss some properties of Lipschitzian elements, which are elements $T$ of $\C_p$ defined by a certain metric condition that allows one to integrate Lipschitzian functions along the Galois orbit of $T$ over $K$ with respect to the Haar distribution.

We strengthen a characterization of the $p$-adic binomial series and a special case of a formal analogue of Hilbert's Theorem 90 for $p$-adic power series.

Let $p$ be an odd prime number, $K$ a $p$-adic field of degree $r$ over ${{\mathbf{Q}}_{p}}$, $O$ the ring of integers in $K,\,B\,=\,\{{{\beta }_{1}},\ldots .{{\beta }_{r}}\}$ an integral basis of $K$ over ${{\mathbf{Q}}_{p}}$, $u$ a unit in $O$ and consider sets of the form $N\,=\,\{{{n}_{1}}{{\beta }_{1}}\,+\ldots +\,{{n}_{r}}{{\beta }_{r}}\,:\,1\,\le \,{{n}_{j}}\,\le \,{{N}_{j}},\,1\,\le \,j\,\le \,r\}$. We show under certain growth conditions that the pair correlation of $\{u{{z}^{2}}\,:\,z\,\in N\}$ becomes Poissonian.

The generating degree $\text{g}\deg \left( A \right)$ of a topological commutative ring $A$ with char $A\,=\,0$ is the cardinality of the smallest subset $M$ of $A$ for which the subring $\mathbb{Z}\left[ M \right]$ is dense in $A$. For a prime number $p$, ${{\mathbb{C}}_{p}}$ denotes the topological completion of an algebraic closure of the field ${{\mathbb{Q}}_{p}}$ of $p$-adic numbers. We prove that $\text{g}\deg \left( {{\mathbb{C}}_{p}} \right)\,=\,1$, i.e., there exists $t$ in ${{\mathbb{C}}_{p}}$ such that $\mathbb{Z}\left[ t \right]$ is dense in ${{\mathbb{C}}_{p}}$. We also compute $\text{gdeg}\left( A\left( U \right) \right)$ where $A\left( U \right)$ is the ring of rigid analytic functions defined on a ball $U$ in ${{\mathbb{C}}_{p}}$. If $U$ is a closed ball then $\text{gdeg}\left( A\left( U \right) \right)\,=\,2$ while if $U$ is an open ball then $\text{gdeg}\left( A\left( U \right) \right)$ is infinite. We show more generally that $\text{gdeg}\left( A\left( U \right) \right)$ is finite for any affinoid$U$ in ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$ and $\text{gdeg}\left( A\left( U \right) \right)$ is infinite for any wide open subset $U$ of ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$.

We prove that two, apparently different, class-group valued Galois module structure invariants associated to the algebraic $K$-groups of rings of algebraic integers coincide. This comparison result is particularly important in making explicit calculations.