Geometry & Topology Seminar

Geometry & Topology Seminar 

Date: Wednesday, 23 September 2026 

Time: 12noon-1pm 

Speaker: June Park (The University of Sydney) 

Location: SMRI Seminar Room (Macleay A12 Room 301) 

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

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 $1\leq r\leq \left\lfloor\frac{10n-2}{n-1}\right\rfloor$.