Random Dieudonné modules, random \(p\)-divisible groups, and random curves over finite fields (Q2841764)

Let \(p>0\) be a prime number and \(q\) a power of \(p\). Let \(C\) be a genus-\(g\) hyperelliptic curve over \(\mathbb{F}_q\) defined by \(y^2=f(x)\). Cohen-Lenstra distribution predicts that the probability distribution on the isomorphism classes of \(\mathrm{Cl}(\mathbb{F}_q(C))[\ell^{\infty}]\) approaches to limit as the genus \(g\) goes to infinity. Here \(\ell\) is prime that is different from \(p\). \vskip 0.1in The subject paper studies the case when \(\ell = p\). More precisely, the paper addresses questions such as do invariants (such as \(a\)-number, \(p\)-rank, etc) of the \(p\)-divisible group of \(\mathrm{Jac}(C_f)\) approaches to a limit when the genus \(g\) goes to infinity? If so, what is the limit? \vskip 0.1in By the classical Dieudonné theory, the category of \(p\)-divisible groups over finite field \(k\) is equivalent to the category of Dieudonné modules over \(k\). The paper defines a probability distribution on the isomorphism classes of principally quasi-polarized Dieudonné module \((D,F,V,\omega)\) of rank \(2g\) over \(\mathbb{Z}_q\) by proving that the double coset \(\mathcal{F}(D)\) of a \(p\)-autodual endomorphism of \(D\) inside the group of \(\mathbb{Z}_q\)-linear symplectomorphisms \(\mathrm{Sp}(D)\) actually contains all the \(p\)-autodual \(\sigma\)-linear endomorphisms of \(D\). As a result, \(\mathcal{F}(D)\) is endowed with a probability measure inherited from Haar measure on \(\mathrm{Sp}(D) \times \mathrm{Sp}(D)\). A random principally quasi-polarized Dieudonné module \((D, F, V, \omega)\) of dimension \(2g\) is one that we choose \(F\) randomly from \(\mathcal{F}(D)\). The paper proves probabilities of several invariants of \(p\)-divisible groups including \(a\)-number, \(p\)-corank. We state one here:NEWLINENEWLINEProposition. The probability that a random principally quasi-polarized \(p\)-divisible group of rank \(2g\) over \(\mathbb{F}_q\) has \(a\)-number \(r\) approaches NEWLINE\[NEWLINEq^{-\binom{r+1}{2}} \prod_{i=1}^{\infty}(1+q^{-i})^{-1} \prod_{i=1}^r (1-q^{-i})^{-1},NEWLINE\]NEWLINE as \(g \to \infty\). \vskip 0.2in Let \(\{M_i\}_{i=1}^{\infty}\) be a family of schemes and \(A_i\) be an abelian scheme over \(M_i\). In the second part of the paper, the authors discuss numerical evidence concerning the question: which of these families, with respect to to which statistics \(X\), yield random \(p\)-divisible groups in the following sense NEWLINE\[NEWLINE\lim_{g \to \infty} \frac{\sum_{y \in M_g(\mathbb{F}_q)} X(A_{g,y}[p^{\infty}])}{|M_g(\mathbb{F}_q)|} = \mathcal{E}(X)?NEWLINE\]NEWLINENEWLINENEWLINEThe authors suggest some relationship between the heuristics presented in the paper and the cohomology of moduli spaces of curves and abelian varieties in positive characteristic. Some tables of data are also presented and the Magma code that performs the computations can be found at the third author's web page.