Class groups and Selmer groups

By Edward F. Schaefer. 2015

77 citations recorded by the source corpus.

It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is to their intersection. In this paper we compute the two groups and their intersection explicitly in the local case and put together the local information to get sharp upper bounds in the global case. The techniques in this paper can be used for arbitrary abelian varieties, isogenies and number fields assuming a frequently occurring condition. Several examples are worked out for the Jacobians of elliptic and hyperelliptic curves.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/08d6a079c36e8927a59f683b9c780fad0258f3f8

How researchers see this paper

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In Algebra and Number Theory

1. Tim Dokchitser
2. Vladimir Dokchitser
3. Franz Lemmermeyer
4. Manjul Bhargava
5. A. Shankar

In Computational Theory and Mathematics

1. Vladimir Dokchitser
2. Franz Lemmermeyer
3. Tim Dokchitser

In Geometry and Topology

1. Tim Dokchitser
2. Vladimir Dokchitser
3. Franz Lemmermeyer
4. Manjul Bhargava
5. A. Shankar

Citations in the corpus

• Local invariants of isogenous elliptic curves (2012, background): “ c . Proof. (1) By the N´eron universal property, φ extends to a morphism of N´eron models, and thus induces a map on formal groups. For the leading term, see [22] Ch. IV, especially Cor. IV.4.3. (2) [18] Lemma 3.8. Lemma 4.3. If l 6= p, then φ∗ω′ is minimal, so α φ/K = 1. Proof. Write φ∗ω′ = aω, (φ”
• Root numbers and parity of ranks of elliptic curves (2009, unknown): “it is spelled out for abelian varieties in [23] Lemma 3.”
• Class Field Towers (2010, methodology): “Washington’s article presents connections of the 2-class fields with certain elliptic curves; see also Eisenbeis, Frey and Ommerborn [346], Nakano [78], Kawachi and Nakano [363], U. Schneiders [368, 369, 374] and Schaefer’s paper [371], where he constructs a noncyclic cubic field whose 2-class group”