Quadratic spaces and Selmer groups of abelian varieties with multiplication

2025/04/30 by Shu, Jie · 1 citation
#11G05 #14K15 #14K22 #FOS: Mathematics #Number Theory (math.NT) #Primiary 11G10 #Secondary 11E04

paper · doi:10.48550/arxiv.2504.21272

Abstract

For certain symmetric isogeny λ: A→ A^\vee of abelian varieties over a global field F, B. Poonen and E. Rains put an orthogonal quadratic structure on H¹(\mathbbA_F,A[λ]) and realize the Selmer group Sel_λ(A) as an intersection of two maximal isotropic subspaces of H¹(\mathbbA_F,A[λ]). With this understanding of Selmer groups, they expect to model the Selmer groups of elliptic curves and Jacobian varieties of hyperelliptic curves as the intersections of random maximal isotropic subspaces of orthogonal spaces. We extend this phenomenon to abelian varieties with multiplication and discuss the Shafarevich-Tate groups.

Citations

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Cited by

• Selmer ranks in twists of CM abelian varieties

Related

• On the λ-invariant of Selmer groups arising from certain quadratic twists of Gross curves
• Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields
• Growth of Selmer Groups over function fields
• Le groupe de Selmer des isogénies de hauteur un
• Selmer groups of abelian varieties in extensions of function fields