Finding large Selmer rank via an arithmetic theory of local constants. (English) Zbl 1219.11084

Summary: We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields.
Suppose \(K/k\) is a quadratic extension of number fields, \(E\) is an elliptic curve defined over \(k\), and \(p\) is an odd prime. Let \(\mathcal K^-\) denote the maximal abelian \(p\)-extension of \(K\) that is unramified at all primes where \(E\) has bad reduction and that is Galois over \(k\) with dihedral Galois group (i.e., the generator \(c\) of \(\text{Gal}(K/k)\) acts on Gal\((\mathcal K^-/K)\) by inversion). We prove (under mild hypotheses on \(p\)) that if the \(\mathbb Z_p\)-rank of the pro-\(p\) Selmer group \(\mathcal S_p(E/K)\) is odd, then rank\(_{\mathbb Z_p} \mathcal S_p(E/F) \geq [F:K]\) for every finite extension \(F\) of \(K\) in \(\mathcal K^-\).


11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R23 Iwasawa theory
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