On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

Castella, Francesc; Grossi, Giada; Lee, Jae‐Hoon; Skinner, Christopher

doi:10.1007/s00222-021-01072-y

Cited by 12 publications

(16 citation statements)

References 44 publications

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“…As well as in other results on the p-converse theorem in rank 1 without a finiteness condition on the Tate-Shafarevich group that appeared after[49]:[18,21,51], etc.…”

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confidence: 52%

$$p^\infty $$-Selmer groups and rational points on CM elliptic curves

et al. 2022

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R\'esum\'eLet $$E/{\mathbb {Q}}$$ E / Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if $$\text {Sel}_{p^\infty }(E/{\mathbb {Q}})$$ Sel p ∞ ( E / Q ) has $${\mathbb {Z}}_p$$ Z p -corank one, then $$E({\mathbb {Q}})$$ E ( Q ) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus $${{\,\mathrm{ord}\,}}_{s=1}L(E,s)=1$$ ord s = 1 L ( E , s ) = 1 , yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For $$p>3$$ p > 3 , this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].

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“…As well as in other results on the p-converse theorem in rank 1 without a finiteness condition on the Tate-Shafarevich group that appeared after[49]:[18,21,51], etc.…”

supporting

confidence: 52%

$$p^\infty $$-Selmer groups and rational points on CM elliptic curves

et al. 2022

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“…Similarly, the prior results in the rank zero case (cf. [26], [15], [39], [9]) exclude the twists covered by Theorems 3.3 and 3.5 and Corollary 3.3.…”

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confidence: 94%

“…Theorems 3.1 and 3.2 provide many families of twists that complement the prior results towards the Birch and Swinnerton-Dyer formula for elliptic curves with analytic rank one (cf. [44], [19], [9], [8]). Notice in particular that [9] treats a general yet completely complementary Eisenstein case: ϕψ being either ramified at p and odd, or unramified at p and even.…”

Section: Supplementamentioning

confidence: 99%

A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes

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Skinner

2023

Proc. Amer. Math. Soc. Ser. B

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We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ E_{\psi } of E E have Mordell–Weil rank one and the 3 3 -adic height pairing on E ψ ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero.

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“…[44], [19], [9], [8]). Notice in particular that [9] treats a general yet completely complementary Eisenstein case: ϕψ being either ramified at p and odd, or unramified at p and even. Similarly, the prior results in the rank zero case (cf.…”

Section: Supplementamentioning

confidence: 99%

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2023

Proc. Amer. Math. Soc. Ser. B

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We show that for certain non-CM elliptic curves E /Q such that 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ of E have Mordell-Weil rank one and the 3-adic height pairing on E ψ (Q) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p-adic height. It is not known -though expected -that the archimedian height of these higher-codimensional cycles is non-zero.

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On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

Cited by 12 publications

References 44 publications

$$p^\infty $$-Selmer groups and rational points on CM elliptic curves

$$p^\infty $$-Selmer groups and rational points on CM elliptic curves

A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes

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