Uniform Manin-Mumford for a family of genus 2 curves

By Laura Demarco, Holly Krieger, Hexi Ye. Annals of Mathematics, 2019

58 citations recorded by the source corpus.

We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb{C}$, and a uniform Bogomolov bound for the family over $\overline{\mathbb{Q}}.$

Subfields: Geometry and Topology, Mechanics of Materials, Sociology and Political Science

View Paper: https://www.semanticscholar.org/paper/80c7d7a8596827a13bc56ae3c600bba0a7c3db69

Citations in the corpus

• Uniformity in Mordell–Lang for curves (2020, unknown): “4, DeMarco–Krieger– Ye [DKY20] proved a bound on the cardinality of torsion points on any genus 2 curve that admits a degree-two map to an elliptic curve when the Abel–Jacobi map is based at a Weierstrass point.”
• Equidistribution in Families of Abelian Varieties and Uniformity (2021, unknown): “Whereas the number of torsion points on ι D ( C ) ⊆ Jac( C ) is uniformly bounded by the theorem, there can be no uniform bound on their order, even under the additional restriction D = [ p ] with a closed point p ∈ C (see [16, Section 9.5] for a counterexample to such a strengthening).”