Local values of G-functions at CM points

Abstract

We study G-functions attached to one--parameter families of elliptic curves centered at a CM point. Using de Rham--Betti and de Rham--crystalline period matrices, we construct non-trivial algebraic relations among their values at CM points at every place of proximity, extending work of André and Beukers. These relations yield effective height bounds via the André--Bombieri Hasse principle. We also discuss a connection with class number bounds for imaginary quadratic fields.

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