Finding large Selmer rank via an arithmetic theory of local constants

2007/09/01 by Barry Mazur, Karl Rubin · 81 citations
Computer Science · Mathematics · #Abelian extension #Abelian group #Algebraic Geometry and Number Theory #Algebraic number field #Analytic Number Theory Research #Combinatorics #Cryptography and Residue Arithmetic #Cyclic group #Dihedral angle #Discrete mathematics #Elliptic curve #Galois extension #Galois group #Galois module #Genus field #Mathematics #Prime (order theory) #Pure mathematics #Rank (graph theory)

paper · pdf · doi:10.4007/annals.2007.166.579

published in Annals of Mathematics 166(2), 579-612 (Princeton University)

openalex publication_date 2007/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields.

Cited by

• Disparity in Selmer ranks of quadratic twists of elliptic curves
• Strong Selmer companion elliptic curves
• Twisting commutative algebraic groups
• Growth of Selmer Groups over function fields
• The Cassels-Tate pairing for finite Galois modules
• p-twisted Selmer near-companion curves
• Selmer groups of elliptic curves in degree p extensions
• A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists
• On refined conjectures of Birch and Swinnerton-Dyer type for\n Hasse-Weil-Artin L-series
• Selmer groups as flat cohomology groups
• On 2-Selmer ranks of quadratic twists of elliptic curves
• Twisting commutative algebraic groups
• On the Galois structure of Selmer groups
• The ℓ-parity conjecture over the constant quadratic extension
• Growth of Selmer rank in nonabelian extensions of number fields
• On the Iwasawa Main conjecture of abelian varieties over function fields
• Ranks of twists of elliptic curves and Hilbert’s tenth problem
• Selmer ranks in twists of CM abelian varieties
• Quadratic spaces and Selmer groups of abelian varieties with multiplication
• On the parity of ranks of Selmer groups IV. With an appendix by Jean-Pierre Wintenberger
• On the Birch-Swinnerton-Dyer quotients modulo squares
• p-Selmer growth in extensions of degree p

Related

• Finding large Selmer rank via an arithmetic theory of local constants
• On the parity of ranks of Selmer groups IV. With an appendix by Jean-Pierre Wintenberger
• On 2-Selmer ranks of quadratic twists of elliptic curves
• A heuristic for boundedness of ranks of elliptic curves
• Elliptic curves with long arithmetic progressions have large rank