Mazur’s conjecture on higher Heegner points

By C. Cornut. 2002

137 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/8230edee78254d9a9bfd54ef4d18d6112d463dc2

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Citations in the corpus

Finding Large Selmer Rank via an Arithmetic Theory of Local Constants (2005, background): “anticyclotomic Zp-extension of K, and thanks to [ Co ] and [V] we know that the bulk of the contribution to the Selmer groups in Theorem 8.3 comes from Heegner points.”
On anticyclotomic μ-invariants of modular forms (2006, background): “If N has an even number of prime divisors, then it is known by work of Cornut [8] and Vatsal [32] that E has infinitely many independent non-torsion points defined over ring class fields of K of p-power conductor.”
L -functions and Galois Representations: Nontriviality of Rankin-Selberg L -functions and CM points (2007, background): “We note however that everything works with the obvious modifications in the non-compact case as well – see [4] and [27], as well as [5].”
Subvarieties of Shimura varieties (2001, background): “For example Vatsal and Cornut (see [19] and [5]) consider sets of Heegner points on products of modular curves where the discriminants of the endomorphism rings of the elliptic curves in question are products of a fixed finite set of prime numbers (of course, the André-Oort conjecture in this case w”
A heuristic for boundedness of ranks of elliptic curves (2016, background): “s of N split in K0. Let K∞ be the anticyclotomic Zp-extension of K0, let Kn be the degree pn subextension, and let Λ := Zp[[Gal(K∞/K)]] be theIwasawa algebra[Maz84, Section 17]. Thetheorem onpage496of[Cor02]implies Mazur’s conjecture [Maz84, bottom of p. 203] that the Heegner module E(K∞) [Maz84, p.”
Ranks of elliptic curves (2002, result): “What Nekovář proved (using recent results of Vatsal [74] and Cornut [10]) is that if p is a prime not dividing #E(Q)tors, and if E has good ordinary reduction at p (see Chapters V and VII of [68]), then dimFp Sp(E) and rankan(E) have the same parity.”
The Heegner point Kolyvagin system (2004, result): “By the main result of [Cor02], one of the points NormKk[1]/KkPk[1] has infinite order, and so Proposition 10 of section 3 of [PR87] implies that H is free of rank one.”
Variation of Heegner points in Hida families (2006, methodology): “The precise result, Corollary 3.1.2, is actually somewhat stronger and extends Cornut and Vatsal’s proof of Mazur’s conjecture on the nonvanishing of Heegner points from the case of modular forms of weight two and trivial character to all ordinary modular forms.”
Special values of anticyclotomic $L$-functions (2003, methodology): “nonzero in E(Hn) Q() and C , respectively, for all but finitely many .”
Selmer groups and the indivisibility of Heegner points (2014, unknown): “[43, 4, 9] and also §6).”
CM points and quaternion algebras (2005, result): “Although the statements are somewhat complicated, they are not hard to prove, in view of our earlier results [2], [18], [19], where all the main ideas are already present.”