On the conjecture of Mazur, Tate, and Teitelbaum

Greenberg, Ralph; Stevens, Glenn

doi:10.1090/conm/165/01607

Cited by 14 publications

(11 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The periods Ω ± f E are the real and imaginary periods associated to the newform f E constructed by Shimura ([51]). See also Theorem 3.5.4 in the work of Greenberg-Stevens [19]. These periods are well defined up to units in Z is a p-adic unit.…”

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

confidence: 91%

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

Lei

Palvannan

2019

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

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

show abstract

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

confidence: 91%

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

Lei

Palvannan

2019

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

“…We assume that F = Q (otherwise the theorem is known by [H04] and by Greenberg-Stevens [GS] and [GS1]). Take first a quaternion algebra…”

Section: Hecke Algebras For Quaternion Algebrasmentioning

confidence: 99%

L-invariants of Tate Curves

Hida¹

2009

Pure and Applied Mathematics Quarterly

View full text Add to dashboard Cite

“…Such zeros, called trivial, were first considered for an elliptic curve E over Q in the seminal work of Mazur, Tate and Teitelbaum [36]. If E has split multiplicative reduction at p, the p-adic L-function L p (E, s) has a trivial zero at s = 1 and it was conjectured, and later proven by Greenberg and Stevens [27], that…”

Section: Introductionmentioning

confidence: 99%

“…The conjectural non-vanishing of the L -invariant is currently only known for elliptic curves over Q (see [1]). Previously, the Trivial Zero Conjecture at the central point was proved for modular forms over Q in [27,47], and for parallel weight 2 ordinary Hilbert cusp forms in [37,46], the latter building on ideas of [21,38]. The non-criticality condition is implied by the assumption of π having non-critical slope (see Prop.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture

Barrera¹,

Dimitrov²,

Jorza³

2017

Preprint

View full text Add to dashboard Cite

We prove a strong form of the trivial zero conjecture at the central point for the p-adic L-function of a non-critically refined cohomological cuspidal automorphic representation of GL2 over a totally real field, which is Iwahori spherical at places above p.In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the p-adic L-function of a nearly finite slope family and on improved p-adic L-functions that we construct using automorphic symbols.To compute higher order derivatives of p-adic L-functions we go beyond the Greenberg-Stevens method and study the variation of the root number in partial finite slope families to establish the vanishing of many Taylor coefficients of the p-adic L-function of the family.

show abstract

On the conjecture of Mazur, Tate, and Teitelbaum

Cited by 14 publications

References 0 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

L-invariants of Tate Curves

$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture

Contact Info

Product

Resources

About