Explicit bounds for split reductions of simple abelian varieties
Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 41-55.

Soit X/K une variété abélienne absolument simple, définie sur un corps de nombres ; nous étudions comment les réductions X 𝔭 tendent à être simples également. Nous montrons que si End(X) est une algèbre de quaternions définie, alors la réduction X 𝔭 est géométriquement isogène au self-produit d’une variété abélienne absolument simple, ce pour 𝔭 dans un ensemble de densité strictement positive, alors que si X est de type Mumford, X 𝔭 est simple pour presque tout 𝔭. Pour une large classe de variétés abéliennes avec anneau d’endomorphismes absolus commutatif, nous donnons une borne supérieure explicite pour la croissance de l’ensemble des premiers de réduction non-simple.

Let X/K be an absolutely simple abelian variety over a number field; we study whether the reductions X 𝔭 tend to be simple, too. We show that if End(X) is a definite quaternion algebra, then the reduction X 𝔭 is geometrically isogenous to the self-product of an absolutely simple abelian variety for 𝔭 in a set of positive density, while if X is of Mumford type, then X 𝔭 is simple for almost all 𝔭. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound for the growth of the set of primes of non-simple reduction.

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DOI : https://doi.org/10.5802/jtnb.787
Classification : 11G25
@article{JTNB_2012__24_1_41_0,
     author = {Jeffrey D. Achter},
     title = {Explicit bounds for split reductions of simple abelian varieties},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {41--55},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {1},
     year = {2012},
     doi = {10.5802/jtnb.787},
     zbl = {pre06075021},
     mrnumber = {2914900},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.787/}
}
Jeffrey D. Achter. Explicit bounds for split reductions of simple abelian varieties. Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 41-55. doi : 10.5802/jtnb.787. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.787/

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