The Analytic Subgroup Theorem

It is well-known that the Wustholz’ analytic subgroup theorem is one of the most powerful theorems in transcendence theory. The theorem gives in a very systematic and conceptual way the transcendence of a large class of complex numbers, e.g. the transcendence of π which is originally due to Lindemann. In this paper we revisit the p-adic analogue of the analytic subgroup theorem and present a proof based on the method described and developed by the authors in a recent related paper.

Non-commutative analytic subgroup theorem

We use a $p$-adic analogue of the analytic subgroup theorem of Wustholz to deduce the transcendence and linear independence of some new classes of $p$-adic numbers. In particular we give $p$-adic analogues of results of Wustholz contained in [G. Wustholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkhauser Boston, Boston, MA, (1983), 329-336] and generalizations of results obtained by Bertrand in [D. Bertrand, Sous-groupes a un parametre $p$-adique de varietes de groupe, Invent. Math. 40 (1977), no. 2, 171-193] and [D. Bertrand, Problemes locaux, Societe Mathematique de France, Asterisque 60-70 (1979), 163-189].

We use the $$p$$ -adic analytic subgroup theorem to give new proofs of some transcendence theorems on the values of $$p$$ -adic exponential and $$p$$ -adic elliptic functions.

Exceptional sets of hypergeometric series

In this paper, we introduce the notion of a bi- \overline{\mathbb{Q}} -structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi- \overline{\mathbb{Q}} -structure decomposes into the direct sum of 1 -dimensional bi- \overline{\mathbb{Q}} -subspaces, and make this decomposition explicit for the moduli space of abelian varieties \mathbb{A}_{g} .We propose an analytic subspace conjecture , which is the analogue of the Wüstholz’s analytic subgroup theorem in this context. We show that this conjecture, applied to \mathbb{A}_{g} , implies that all quadratic \overline{\mathbb{Q}} -relations among the holomorphic periods of CM abelian varieties arise from elementary ones.

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.

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