Chantal David (Concordia).
Let Gm,k := Z/mZ
× Z/mkZ be an abelian group of rank 2 and order N = mk
does there exist a ﬁnite ﬁeld Fp and an elliptic curve E/Fp such that E(Fp )
≃ Gm,k ? We
show that this happens with probability 0 when k is very small with respect to m, and
with probability 1 when k is big enough with respect to m. The fact that the groups Gm,k
are more likely to occur when k is big is reminiscent of the Cohen-Lenstra heuristics which
predict that a random abelian group G occurs with probability weighted by #G/#Aut(G).
By counting the average number of times that a given group Gm,k occurs over the ﬁnite
ﬁelds Fp (and not simply when a given group occurs or not), we are able to verify that
the probability of occurrence of the groups Gm,k is indeed weighted by the Cohen-Lenstra
This is joint work with V. Chandee, D. Koukoulopoulos and E. Smith.