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In this paper, we formulate and prove the so-called $p$-adic non-commutative analytic subgroup theorem. This result is seen as the $p$-adic analogue of a recent theorem given by Yafaev in [11].

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]. [ABSTRACT FROM AUTHOR]

In this paper, we give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and p-adic fields, respectively. [ABSTRACT FROM AUTHOR]

Given $E \subseteq \mathbb {F}_q^d \times \mathbb {F}_q^d$ , with the finite field $\mathbb {F}_q$ of order q and the integer $d\,\ge \, 2$ , we define the two-parameter distance set $\Delta _{d, d}(E)=\{(\|x-y\|, \|z-t\|) : (x, z), (y, t) \in E \}$. Birklbauer and Iosevich ['A two-parameter finite field Erdős–Falconer distance problem', Bull. Hellenic Math. Soc. 61 (2017), 21–30] proved that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}(E)| = q^2$. For $d=2$ , they showed that if $|E| \gg q^{{10}/{3}}$ , then $ |\Delta _{2, 2}(E)| \gg q^2$. In this paper, we give extensions and improvements of these results. Given the diagonal polynomial $P(x)=\sum _{i=1}^da_ix_i^s\in \mathbb F_q[x_1,\ldots , x_d]$ , the distance induced by P over $\mathbb {F}_q^d$ is $\|x-y\|_s:=P(x-y)$ , with the corresponding distance set $\Delta ^s_{d, d}(E)=\{(\|x-y\|_s, \|z-t\|_s) : (x, z), (y, t) \in E \}$. We show that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}^s(E)| \gg q^2$. For $d=2$ and the Euclidean distance, we improve the former result over prime fields by showing that $ |\Delta _{2,2}(E)| \gg p^2$ for $|E| \gg p^{{13}/{4}}$. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Schmidt's subspace theorem for non-subdegenerate families of hyperplanes. In particular, our result improves the previous result on Schmidt's subspace type theorem for the case of non-degenerate families of hyperplanes, and furthermore, also shows the sharpness of the condition of non-subdegeneracy. As a consequence, we deduce a version of Lang's conjecture on exceptional sets in the case of complements of hyperplanes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and prove the so-called p-adic non-commutative analytic subgroup theorem. This result is seen as the p-adic analogue of a recent theorem given by Yafaev in [11]. [ABSTRACT FROM AUTHOR]