Rankin–Eisenstein classes in Coleman families

Loeffler, David; Zerbes, Sarah Livia

doi:10.1186/s40687-016-0077-6

Cited by 50 publications

(105 citation statements)

References 28 publications

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“…] . For the reverse inequality, one can employ an argument involving Euler systems of Beilinson-Flach elements constructed by Loeffler-Zerbes in [34]. Under some technical hypotheses, such an Euler system is constructed and used to prove this direction of the main conjecture in [ For the remainder of this section, it will be helpful to keep the following picture in mind:…”

Section: Two-variable Main Conjectures and Two Variable P-adic L-funcmentioning

confidence: 99%

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Lei

Palvannan

2019

Forum of Mathematics, Sigma

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A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's 2-variable p-adic L-functions) and algebraic objects (two "everywhere unramified" Iwasawa modules) involving codimension two cycles in a 2-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field K (where an odd prime p splits) of an elliptic curve E, defined over Q, with good supersingular reduction at p. On the analytic side, we consider eight pairs of 2-variable p-adic L-functions in this setup (four of the 2-variable p-adic L-functions have been constructed by Loeffler and a fifth 2-variable p-adic L-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the Z 2 p -extension of K. We also provide numerical evidence, using algorithms of Pollack, towards a pseudo-nullity conjecture of Coates-Sujatha.1 The first subscript d of ρ d,n indicates the dimension of the Galois representation, while the second subscript n denotes a number one less than the Krull dimension of the ring R. In the settings we are interested in, the number n would denote the number of variables in the corresponding p-adic L-functions.2 The Panchishkin condition is a type of "ordinariness" assumption, introduced by Greenberg, while formulating the Iwasawa main conjecture for Galois deformations. See Section 4 in [14] for the precise definition. 2

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Section: Two-variable Main Conjectures and Two Variable P-adic L-funcmentioning

confidence: 99%

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Lei

Palvannan

2019

Forum of Mathematics, Sigma

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“…Their approach also uses syntomic cohomology to obtain the result for many specialisations of the family, and analytic continuation to obtain the result everywhere; but there is a significant difference between their proof and ours, in that they analytically continue from specialisations of weight 2 and high p-power level, rather than high weight and prime-to-p level as in our approach. Thus their strategy requires a delicate study of the special fibres of the modular curves X 1 (N p r ) in characteristic p, which is not needed in our approach; and our method is also amenable to generalisations to non-ordinary Coleman families, as we shall show in a subsequent paper [LZ16].…”

Section: Introductionmentioning

confidence: 99%

Rankin–Eisenstein classes and explicit reciprocity laws

Kings

Loeffler

Zerbes

2017

Cambridge Journal of Mathematics

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183

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Abstract. We construct three-variable p-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated L-value does not vanish.

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“…Moreover we remark that our methods are amenable to being generalized to Hilbert modular forms. This is a big difference with Kato's Euler system used in [22] or the Euler system of Rankin-Eisenstein classes used in [23,24].…”

Section: Introductionmentioning

confidence: 99%

Selmer groups and central values of L-functions for modular forms

Chida¹

2017

Ann. inst. Fourier

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Rankin–Eisenstein classes in Coleman families

Cited by 50 publications

References 28 publications

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Rankin–Eisenstein classes and explicit reciprocity laws

Selmer groups and central values of L-functions for modular forms

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