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Originalsprache | undefiniert/unbekannt |
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Publikationsstatus | Elektronisch veröffentlicht (E-Pub) - 20 Feb. 2024 |
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2024.
Publikation: Arbeitspapier/Preprint › Preprint
}
TY - UNPB
T1 - On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction
AU - Keller, Timo
AU - Yin, Mulun
N1 - Comments welcome
PY - 2024/2/20
Y1 - 2024/2/20
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.
KW - math.NT
KW - 11G40 (Primary) 11G05, 11G10, 14G10 (Secondary)
M3 - Preprint
BT - On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction
ER -