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On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction

Publikation: Arbeitspapier/PreprintPreprint

Autorschaft

  • Timo Keller
  • Mulun Yin

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Originalspracheundefiniert/unbekannt
PublikationsstatusElektronisch veröffentlicht (E-Pub) - 20 Feb. 2024

Abstract

Let f be a newform of weight k and level N with trivial nebentypus. Let \mathfrak{p}\nmid 2N be a maximal prime ideal of the coefficient ring of f such that the self-dual twist of the mod-\mathfrak{p} Galois representation of f is reducible with constituents \phi,\psi. Denote a decomposition group over the rational prime p below \mathfrak{p} by G_p. We remove the condition \phi|_{G_p} \neq \mathbf{1}, \omega from [CGLS22], and generalize their results to newforms of arbitrary weights. As a consequence, we prove some Iwasawa main conjectures and get the p-part of the strong BSD conjecture for elliptic curves of analytic rank 0 or 1 over \mathbf{Q} in this setting. In particular, non-trivial p-torsion is allowed in the Mordell--Weil group. Using Hida families, we prove a Iwasawa main conjecture for newforms of weight 2 of multiplicative reduction at Eisenstein primes. In the above situations, we also get p-converse theorems to the theorems of Gross--Zagier--Kolyvagin. The p-converse theorems have applications to Goldfeld's conjecture in certain quadratic twist families of elliptic curves having a 3-isogeny.

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On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction. / Keller, Timo; Yin, Mulun.
2024.

Publikation: Arbeitspapier/PreprintPreprint

Keller, T., & Yin, M. (2024). On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction. Vorabveröffentlichung online.
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N2 - Let f be a newform of weight k and level N with trivial nebentypus. Let \mathfrak{p}\nmid 2N be a maximal prime ideal of the coefficient ring of f such that the self-dual twist of the mod-\mathfrak{p} Galois representation of f is reducible with constituents \phi,\psi. Denote a decomposition group over the rational prime p below \mathfrak{p} by G_p. We remove the condition \phi|_{G_p} \neq \mathbf{1}, \omega from [CGLS22], and generalize their results to newforms of arbitrary weights. As a consequence, we prove some Iwasawa main conjectures and get the p-part of the strong BSD conjecture for elliptic curves of analytic rank 0 or 1 over \mathbf{Q} in this setting. In particular, non-trivial p-torsion is allowed in the Mordell--Weil group. Using Hida families, we prove a Iwasawa main conjecture for newforms of weight 2 of multiplicative reduction at Eisenstein primes. In the above situations, we also get p-converse theorems to the theorems of Gross--Zagier--Kolyvagin. The p-converse theorems have applications to Goldfeld's conjecture in certain quadratic twist families of elliptic curves having a 3-isogeny.

AB - Let f be a newform of weight k and level N with trivial nebentypus. Let \mathfrak{p}\nmid 2N be a maximal prime ideal of the coefficient ring of f such that the self-dual twist of the mod-\mathfrak{p} Galois representation of f is reducible with constituents \phi,\psi. Denote a decomposition group over the rational prime p below \mathfrak{p} by G_p. We remove the condition \phi|_{G_p} \neq \mathbf{1}, \omega from [CGLS22], and generalize their results to newforms of arbitrary weights. As a consequence, we prove some Iwasawa main conjectures and get the p-part of the strong BSD conjecture for elliptic curves of analytic rank 0 or 1 over \mathbf{Q} in this setting. In particular, non-trivial p-torsion is allowed in the Mordell--Weil group. Using Hida families, we prove a Iwasawa main conjecture for newforms of weight 2 of multiplicative reduction at Eisenstein primes. In the above situations, we also get p-converse theorems to the theorems of Gross--Zagier--Kolyvagin. The p-converse theorems have applications to Goldfeld's conjecture in certain quadratic twist families of elliptic curves having a 3-isogeny.

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