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The groups of points on abelian varieties over finite fields

Sergey Rybakov (2010)

Open Mathematics

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).

The p -rank stratification of Artin-Schreier curves

Rachel Pries, Hui June Zhu (2012)

Annales de l’institut Fourier

We study a moduli space 𝒜𝒮 g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p . We study the stratification of 𝒜𝒮 g by p -rank into strata 𝒜𝒮 g . s of Artin-Schreier curves of genus g with p -rank exactly s . We enumerate the irreducible components of 𝒜𝒮 g , s and find their dimensions. As an application, when p = 2 , we prove that every irreducible component of the moduli space of hyperelliptic k -curves with genus g and 2 -rank s has dimension g - 1 + s . We also determine all pairs ( p , g ) for...

Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

David Masser, Umberto Zannier (2015)

Journal of the European Mathematical Society

In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...

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Davide Lombardo (0)

Annales de l’institut Fourier

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