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Simultaneous supersingular reductions of CM elliptic curves
We study the simultaneous reductions at several supersingular primes of
elliptic curves with complex multiplication. We show -- under additional
congruence assumptions on the CM order -- that the reductions are surjective
(and even become equidistributed) on the product of supersingular loci when the
discriminant of the order becomes large. This variant of the equidistribution
theorems of Duke and Cornut-Vatsal is an(other) application of the recent work
of Einsiedler and Lindenstrauss on the classification of joinings of
higher-rank diagonalizable actions.Comment: 46 pages. Revised according to the referee's comment
The prime divisors of the number of points on abelian varieties
Let A,A' be elliptic curves or abelian varieties fully of type GSp defined
over a number field K. This includes principally polarized abelian varieties
with geometric endomorphism ring Z and dimension 2 or odd. We compare the
number of points on the reductions of the two varieties. We prove that A and A'
are K-isogenous if the following condition holds for a density-one set of
primes p of K: the prime numbers dividing #A(k_p) also divide #A'(k_p). We
generalize this statement to some extent for products of such varieties. This
refines results of Hall and Perucca (2011) and of Ratazzi (2012).Comment: results generalized; to appear in Journal de Theorie des Nombres de
Bordeau
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