Random maximal isotropic subspaces and Selmer groups

By Bjorn Poonen, Eric M. Rains. 2010

118 citations recorded by the source corpus.

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_p. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable. We then prove that the p-Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over F_p. By modeling the first subspace as being random, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunay's heuristics for p-torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1/2. Many of our results generalize to abelian varieties over global fields.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3443b59bf69f946c2f30b30c083b5f43e458df21

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, methodology): “More recently, by a completely different approach, Poonen and Rains [33] model the Selmer group as a random intersection of isotropic subspaces of a quadratic space, and again, they predict that the average size of the 2-Selmer group should be 3.”
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, background): “One can show that the resulting map η : H(1)(k,D) → H(1)(k, SO(W )) ∼= Z/2Z is an even quadratic form whose associated bilinear form is the cup product on H(1)(k, J [2]) induced from the Weil pairing J [2]×J [2] → μ2 (see [ 51], [38]).”
Most odd degree hyperelliptic curves have only one rational point (2013, methodology): “For each g ≥ 1 and prime p , we may now formulate the following equidis-tribution conjecture, compatible with both the heuristics in [PR12] and the theorems for p = 2 in [BG13].”