Elliptic curves over \(k(t)\), \(\mathbb F_q(t)\), \(\mathbb C(t)\): Height moduli, exact counts, rank jumps

Jun Yong Park
The University of Sydney

Abstract

The notion of height does more than order the infinite set of elliptic curves: it turns their totality into a geometric space. After a brief motivation over \(\mathbb Q\), I will explain this principle over three rational function fields. Over \(k(t)\), an elliptic curve can be viewed as an elliptic surface over \(\mathbb P^1_k\), and its height is the degree of its classifying map to \(\overline{\mathcal M}_{1,1}\). The resulting height moduli stack is of finite type and parametrizes elliptic curves of fixed Faltings height \(n\), with a natural correspondence between its strata and their Kodaira fibre configurations. Over \(\mathbb F_q(t)\), motivic identities in the Grothendieck ring of stacks specialize to exact counts of elliptic curves of bounded height in every characteristic, including \(p=2\) and \(p=3\). Every lower-order term can be traced to a special stratum, automorphism locus, or minimality defect. Over \(\mathbb C(t)\), fixing a Kodaira fibre stratum and applying the Shioda-Tate formula identifies Mordell-Weil rank jumps with additional Hodge classes. Special cycle modularity organizes their possible height pairings, while a separate period map argument produces analytically dense rank jumps throughout the expected transverse range of $ 1\leq r\leq \left\lfloor\frac{10n-2}{n-1}\right\rfloor$.