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The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I
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abstract
In this paper and its sequel, we develop a technique for finding the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields. Given an elliptic curve E over a number field satisfying certain technical conditions, this technique can be used to show that 100% of the quadratic twists of E have rank at most 1. Given a prime $\ell$ and a number field F not containing $\mu_{2\ell}$, this method also shows that the $\ell^{\infty}$-class groups in the family of degree $\ell$ cyclic extensions of F have a distribution consistent with the Cohen-Lenstra-Gerth heuristics. For this work, we develop the theory of the fixed point Selmer group, which serves as the base layer of the $\ell^{\infty}$-Selmer group. This first paper gives a technique for finding the distribution of $\ell^{\infty}$-Selmer groups in certain families of twists where the fixed point Selmer group is stable. In the sequel paper, we will give a technique for controlling fixed point Selmer groups.
Forward citations
Cited by 5 Pith papers
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Quadratic spaces and Selmer groups of abelian varieties with multiplication
For abelian varieties over global fields with multiplication by an order, the Selmer group is the intersection of two maximal isotropic subspaces in an orthogonal, symplectic, unitary, or split unitary quadratic space.
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On the Distribution of Class Groups of Abelian Extensions
For abelian Galois groups Γ, the p-class group splits into a ramification-forced part with infinite average rank and a conjecturally Cohen-Lenstra distributed part; a weighted moment version is proved over F_q(t).
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Elliptic curves of rank one over number fields
For every number field K and every 3-generic elliptic curve E/K with full rational 2-torsion, there are infinitely many quadratic twists of E with rank exactly 1.
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Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents
For elliptic curves with large 3-torsion Galois image, the Selmer-rank distribution over S3-cubic extensions with a fixed quadratic resolvent is a parity mixture of a universal Markov-chain law, giving a 31.95% lower ...
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Iwasawa theory and ranks of elliptic curves in quadratic twist families
For elliptic curves with vanishing 2-adic mu-invariant and small lambda-invariant, the authors show that many quadratic twists have Selmer corank 1, conditionally yielding rank 1 curves when the Tate-Shafarevich group...
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