Let \(V:=\mathbb{Z}_p^{2n}\) (\(p\) a prime and \(n\in\mathbb{N}\)) be equipped with the standard hyperbolic quadratic form \(Q(x_1,\dots,x_n,y_1,\dots,y_n):=\displaystyle{\sum_{i=1}^n x_iy_i}\) and let \[ \text{OGr}_V(\mathbb{Z}_p):= \{Z\text{\;a \;direct\;summand\;of\;}V\;s.t.\;Q_{|Z}=0\} \] be the set ofisotropic direct summands of \(V\). For any two elements \(Z,W\in \text{OGr}_V(\mathbb{Z}_p)\) one can form the short exact sequence \[ R:=(Z\cap W)\otimes \mathbb{Q}_p/\mathbb{Z}_p \hookrightarrow S:=(Z\otimes \mathbb{Q}_p/\mathbb{Z}_p)\cap(W\otimes \mathbb{Q}_p/\mathbb{Z}_p) \twoheadrightarrow T \quad (\mathrm{Seq}(Z,W)) \] (where \(T\) is the cokernel of the natural map). These sequences (modulo isomorphisms) form a subset of the set \(\mathrm{Seq}(\mathbb{Z}_p)\) of isomorphism classes of short exact sequences of \(\mathbb{Z}_p\)-modules of cofinite type. The authors study the main properties of the sequences of type \(\mathrm{Seq}(Z,W)\) and, in particular, show that, as \(Z\) and \(W\) vary in \(\text{OGr}_V(\mathbb{Z}_p)\), these sequences define a probability distribution \(\mathcal{Q}_{2n}\) which, as \(n\mapsto +\infty\), converges to a discrete probability distribution \(\mathcal{Q}\).
The main goal of the paper is to state (and show motivations for and consequences of) a deep conjecture which uses \(\mathrm{Seq}(Z,W)\) as a model for the well known short exact sequence associated to an elliptic curve \(E\) defined over a global field \(k\), i.e., \[ E(k)\otimes \mathbb{Q}_p/\mathbb{Z}_p \hookrightarrow Sel_{p^\infty}(E) \twoheadrightarrow \text Ш[p^\infty]\qquad\qquad (\mathrm{Seq}(E,k))\,.\] { Conjecture 1.3.} Let \(\mathcal{E}\) be the set of isomorphism classes for all curves defined over \(k\) (ordered by height). For any short exact sequence \(\mathrm{Seq}(Z,W)\), the density of the set \(\{E\in\mathcal{E}\,s.t.\, \mathrm{Seq}(E,k)\simeq \mathrm{Seq}(Z,W)\}\) equals the \(\mathcal{Q}\)-probability of \(\mathrm{Seq}(Z,W)\).
The authors show that in \(\mathrm{Seq}(Z,W)\) the \(\mathbb{Z}_p\)-corank of \(R\) is 0 with probability 1/2 and 1 with probability 1/2, moreover \(T\) is always finite. The conjecture then implies the well known conjectures on the distribution of the rank in \(\mathcal{E}\) and on the finiteness of the Tate-Shafarevich group. The authors also focus on possible models for the Tate-Shafarevich and Selmer group using finite abelian groups or cokernels of random alternating matrices for the first (providing the analogue of the Cassels-Tate pairing for them), and giving some more information on the structure of the second to motivate (at least conjecturally) the search for a model for \(Sel\) inside the set of intersections of direct summands.

Reviewer: Andrea Bandini (Parma)