Finding Large Selmer Rank via an Arithmetic Theory of Local Constants

By Barry Mazur, Karl Rubin. 2005

92 citations recorded by the source corpus.

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 κ - 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 Gal(K/k) acts on Gal(κ-/K) by inversion). We prove (under mild hypotheses on p) that if the Zp-rank of the pro-p Selmer group Sp(E/K) is odd, then rank Zp S p (E/F) > [F: K] for every finite extension F of K in κ - .

Subfields: Algebra and Number Theory, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/cb9d04cd802e30460926a32850f322fe72a713d5

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In Algebra and Number Theory

1. Tim Dokchitser
2. Vladimir Dokchitser
3. Barry Mazur
4. Karl Rubin
5. David Burns
6. Manjul Bhargava
7. A. Shankar
8. Antonio Lei

In Geometry and Topology

1. Tim Dokchitser
2. Vladimir Dokchitser
3. Barry Mazur
4. David Burns
5. Karl Rubin
6. O. Venjakob
7. Manjul Bhargava
8. R. Sujatha

In Information Systems

1. Vladimir Dokchitser
2. Tim Dokchitser
3. Barry Mazur
4. Karl Rubin
5. Jorma Jormakka

Citations in the corpus

• On the Birch-Swinnerton-Dyer quotients modulo squares (2006, result): “As another illustration, we consider dihedral groups to obtain similar results to [ 23 ], e.g.”
• On descent theory and main conjectures in non-commutative Iwasawa theory (2007, background): “If M = h1(A)(1) for an abelian variety A that has good ordinary reduction at p and is such that the Tate–Shafarevich group III(A/F ) of A over F is finite, then it is known that the vanishing of j∗(TΩ(MF , Z[GF/Q])) implies the ‘p-part’ of a Birch– Swinnerton-Dyer type formula (see, for example, [35”
• Ranks of twists of elliptic curves and Hilbert’s tenth problem (2009, background): “2] (the proof given in [15] works even if p = 2, and even if v | ∞).”
• Regulator constants and the parity conjecture (2007, background): “Root numbers and parities of Selmer ranks in the last 3 cases have recently been studied by Mazur–Rubin [ 20 , 21], Hachimori–Venjakob [15] and one of us (V.) [8], Rohrlich [31] and Coates–Fukaya–Kato–Sujatha [3]; see also Greenberg’s preprint [12].”