Class groups and Selmer groups. (English) Zbl 0859.11034

J. Number Theory 56, No. 1, 79-114 (1996); corrigendum ibid. 135, 390 (2014).
In this very readable and carefully written paper, the author studies the descent sequence associated to an isogeny between two abelian varieties defined over a number field. The techniques are Galois cohomological in nature.
Let \(\varphi: A\to A'\) be an isogeny of abelian varieties defined over a number field \(K\). Let \(L\) be the extension of \(K\) obtained by adjoining the points of the kernel \(A[\varphi]\) of \(\varphi\) to \(K\). Under the, frequently valid, assumption that \(A[\varphi]\) is a cohomologically trivial \(\text{Gal} (L/K)\)-module, the \(\varphi\)-Selmer group \(S^\varphi (K,A)\) can be viewed as a subgroup of \(\operatorname{Hom} (\text{Gal} (\overline{L}/L), A[\varphi])\). The author compares the Selmer group with the subgroup of homomorphism \(\text{Gal} (\overline{L}/L)\to A [\varphi]\) that are everywhere unramified. By class field theory, he obtains in this way relations between \(S^\varphi (K,A)\) and the ideal class group of \(K\).
The results are very explicit when \(A\) and \(A'\) are elliptic curves or when \(A=A'\) and \(\varphi=2\). In these cases much work has been done previously and it is enlightening to see this work from the point of view of this paper. The author gives several explicit applications of his methods: he exhibits a cubic number field whose class group has 2-rank at least 13 and he exhibits a curve of genus 2 whose Jacobian has a Mordell Weil group of rank 7.
Reviewer: R.Schoof (Roma)


11G10 Abelian varieties of dimension \(> 1\)
14K02 Isogeny
11R29 Class numbers, class groups, discriminants
11R34 Galois cohomology
11G35 Varieties over global fields
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