Infinitely many genus two and three curves of small fixed positive rank

2025/05/28 by Gajović, Stevan, Park, Sun Woo · 1 citation
#! 11G05 #11G30 #14G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.22470

Abstract

We prove that there exist infinitely many genus two curves over ℚ whose Jacobians have ranks between 0 and 11, and genus three curves over ℚ whose Jacobians have ranks 0, 1, and 2. Furthermore, we consider a generalisation of these results over number fields satisfying some conditions.

Citations

• Constructing Jacobians of rank 1
• There are infinitely many elliptic curves over the rationals of rank 2
• Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field
• Hilbert's tenth problem via additive combinatorics
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• Sylvester's Problem and Mock Heegner Points
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• Elliptic Curves x³ + y³ = k of High Rank

Cited by

• Constructing Jacobians of rank 1

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