On the local divisibility of Heegner points

Gross, Benedict H.; Parson, James

doi:10.1007/978-1-4614-1260-1_11

Cited by 16 publications

(18 citation statements)

References 18 publications

(9 reference statements)

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“…since we assume d K < −4 and hence |O × K | = 2. Now plugging in (27) into (23) when ψ = 1, and (28) into (23) when ψ = 1, we establish the lemma. 7.5.…”

Section: The Katz P-adic L-functionmentioning

confidence: 88%

Goldfeld’s Conjecture and Congruences Between Heegner Points

Kriz

2019

Forum of Mathematics, Sigma

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Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever $E$ has a rational 3-isogeny. We also prove the analogous result for the sextic twists of $j$-invariant 0 curves. For a more general elliptic curve $E$, we show that the number of quadratic twists of $E$ up to twisting discriminant $X$ of analytic rank 0 (respectively 1) is $\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between $p$-adic logarithms of Heegner points and apply it in the special cases $p=3$ and $p=2$ to construct the desired twists explicitly. As a by-product, we also prove the corresponding $p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

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“…since we assume d K < −4 and hence |O × K | = 2. Now plugging in (27) into (23) when ψ = 1, and (28) into (23) when ψ = 1, we establish the lemma. 7.5.…”

Section: The Katz P-adic L-functionmentioning

confidence: 88%

Goldfeld’s Conjecture and Congruences Between Heegner Points

Kriz

2019

Forum of Mathematics, Sigma

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“…Remark 5.2. Condition 5 in Assumption 5.1 is introduced in order to "trivialize" the image of the local Kummer map at primes of bad reduction for E. The reader is referred to, e.g., [18] to see how one could impose suitable local conditions at these primes too. We also expect that Assumption 5.1 could be relaxed by using the methods recently proposed by Nekovář in his work on level raising for Hilbert modular forms of weight two ( [29]), which greatly improves the techniques introduced in [3] and then refined in [24].…”

Section: Choice Of Pmentioning

confidence: 99%

Special values of L-functions and the arithmetic of Darmon points

Longo

Rotger

Vigni

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and \chi is a ring class character such that L(E/K,\chi,1) is not 0 we prove a Gross-Zagier type formula relating Darmon points to a suitably defined algebraic part of L(E/K,\chi,1); this generalizes results of Bertolini, Darmon and Dasgupta to the case of division quaternion algebras and arbitrary characters. Finally, as an application of this formula, assuming the rationality conjectures for Darmon points we obtain vanishing results for Selmer groups of E over extensions of K contained in narrow ring class fields when the analytic rank of E is zero, as predicted by the Birch and Swinnerton-Dyer conjecture

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“…, which is an identification of y=ly-modules (by the canonicity). The claim follows from the y=yl-modules decompositions (6). p…”

Section: The Arithmetic Theory Of Local Constantsmentioning

confidence: 87%

“…Note that A r ; y r ; l r À Á is again regular and unramified, as it follows from the above discussion, but may fail to be polarizable. The decomposition (4) yields canonical y=yl n -module decompositions (see also [6] for a direct definition of the l r -adic Selmer groups):…”

Section: 1])mentioning

confidence: 99%

The Arithmetic Theory of Local Constants for Abelian Varieties

Seveso¹

2012

Rend. Sem. Mat. Univ. Padova

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We present a generalization of the theory of local constants developed by B. Mazur and K. Rubin in order to cover the case of abelian varieties, with emphasis to abelian varieties with real multiplication. Let l be an odd rational prime and let L/K be an abelian l-power extension. Assume that we are given a quadratic extension K/k such that L/k is a dihedral extension and the abelian variety A/k is defined over k and polarizable. This theory can be used to relate the rank of the l-Selmer group of A over K to the rank of the l-Selmer group of A over L.

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On the local divisibility of Heegner points

Cited by 16 publications

References 18 publications

Goldfeld’s Conjecture and Congruences Between Heegner Points

Goldfeld’s Conjecture and Congruences Between Heegner Points

Special values of L-functions and the arithmetic of Darmon points

The Arithmetic Theory of Local Constants for Abelian Varieties

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