The Heegner point Kolyvagin system

By Benjamin Howard. Compositio Mathematica, 2004

105 citations recorded by the source corpus.

In Bull. Soc. Math. France 115 (1987), 399–456, Perrin-Riou formulates a form of the Iwasawa main conjecture which relates Heegner points to the Selmer group of an elliptic curve defined over $\mathbb{Q}$, as one goes up the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field K. Building on the earlier work of Bertolini on this conjecture, and making use of the recent work of Mazur and Rubin on Kolyvagin's theory of Euler systems, we prove one divisibility of Perrin-Riou's conjectured equality. As a consequence, one obtains an upper bound on the rank of the Mordell–Weil group E(K) in terms of Heegner points.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e1e941f3632734ecd90843785d853aa0091cb871

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Citations in the corpus

Finding Large Selmer Rank via an Arithmetic Theory of Local Constants (2005, background): “ular we have dim Fp C = dim Fp CF = dim Fp CG = 1 2 dim Fp B. Since C ∼= H1 F+G(K,W)/H 1 F∩G(K,W) and CF ∼= ⊕v∈SHF1 (Kv,W)/H1F∩G (Kv,W), this proves (i). The proof of (ii) uses an argument of Howard ([Hb] Lemma 1.5.7). We have CF ∩ CG = 0 and CF ⊕ CG = B. If x ∈ H1 F+G(K,W), let xS ∈ C ⊂ B be the lo”
Selmer groups and the indivisibility of Heegner points (2014, unknown): “Following the terminology of [26], suitably modified for the Heegner point setting [19], we call the collection κ∞ a Kolyvagin system.”
Iwasawa theory of Heegner points on abelian varieties of GL2 type (2004, methodology): “a pairing H1 F(E,W) ×H1 F(E,W) → D. A straightforward (if tedious) modiﬁcation of the methods of [6] shows that the resulting pairing is alternating; a similar calculation is done in Theorem 1.4.3 of [9]. Theorem 2.2.2. Suppose we have a set of primes L ⊂ L1 with Le ⊂ L for e ≫ 0. Let M denote the s”
Variation of Heegner points in Hida families (2006, result): “Partial results toward Perrin-Riou’s conjecture have been obtained by Bertolini [1] and the author [12, 13].”