2-selmer groups of hyperelliptic curves with marked points

Ananth N. Shankar*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We consider the family of hyperelliptic curves over Q of fixed genus along with a marked rational Weierstrass point and a marked rational nonWeierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2, and that most such curves have only three rational points. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also consider another related family of curves and study the interplay between these two families. There is a family φ of isogenies between these two families, and we prove that the average size of the φ-Selmer groups is exactly 2.

Original languageEnglish (US)
Pages (from-to)267-304
Number of pages38
JournalTransactions of the American Mathematical Society
Issue number1
StatePublished - 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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