The 2-adic eigencurve at the boundary of weight space

Buzzard, Kevin; Kilford, L. J. P.

doi:10.1112/s0010437x05001314

Cited by 52 publications

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“…In Buzzard-Calegari [3], the following theorem is proved, using similar techniques to those in [16], [11] and [4]: Theorem 6. The slopes of the U 2 operator acting on 2-adic overconvergent modular forms of weight 0 are…”

Section: Previous Work and New Directionsmentioning

confidence: 98%

“…It also uses methods developed by Smithline in his thesis [16] and applied by him in [17], which were then also used in the author's paper [11] and in the paper of the author with Buzzard [4].…”

Section: Previous Work and New Directionsmentioning

confidence: 99%

“…Theorem 5 (Buzzard-Kilford [4], Theorem B). Let k be an integer and let χ be a character of conductor 2 n such that χ(−1) This paper will prove a 5-adic analogue of these results for the specific level Γ 0 (25).…”

Section: Previous Work and New Directionsmentioning

confidence: 99%

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On the slopes of the {U_5} operator acting on overconvergent modular forms

Kilford¹

2008

J. Théor. Nombres Bordeaux

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Section: Previous Work and New Directionsmentioning

confidence: 98%

Section: Previous Work and New Directionsmentioning

confidence: 99%

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On the slopes of the {U_5} operator acting on overconvergent modular forms

Kilford¹

2008

J. Théor. Nombres Bordeaux

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“…The results obtained will be independent of r, since it is known that any overconvergent U -eigenform of finite slope must extend to a function on X 0 (1) ≥p −r for all r < p p+1 (see [Buz03]). The slopes of U are somewhat mysterious; the complete list of slopes is known only for p = 2, tame level 1 and weight 0 by [BC05], and for 2-adic, 3-adic and 5-adic weights near the boundary of weight space by [BK05], [Jac03] and [Kil06] respectively. There are conjectures ( [Buz05], [Cla05]) for a general weight, prime and level, but these appear to be rather inaccessible at present.…”

Section: Computations Of Slopesmentioning

confidence: 99%

Spectral Expansions of Overconvergent Modular Functions

Loeffler

2010

International Mathematics Research Notices

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The main result of this paper is an instance of the conjecture made by Gouvêa and Mazur in [GM95], which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k should be spanned by the finite slope Hecke eigenforms. For N = 1, p = 2 and k = 0 we show that this follows from the combinatorial approach initiated by Emerton [Eme98] and Smithline [Smi00], using the classical LU decomposition and results of ; this implies the conjecture for all r ∈ ( 5 12 , 7 12 ). Similar results follow for p = 3 and p = 5 with the assumption of a plausible conjecture, which would also imply formulae for the slopes analogous to those of [BC05].We also show that (for general p and N ) the space of weight 0 overconvergent forms carries a natural inner product with respect to which the Hecke action is self-adjoint. When N = 1 and p ∈ {2, 3, 5, 7, 13}, combining this with the combinatorial methods allows easy computations of the q-expansions of small slope overconvergent eigenfunctions; as an application we calculate the q-expansions of the first 20 eigenfunctions for p = 5, extending the data given in [GM95]. BackgroundLet S k (Γ 1 (N )) denote the space of classical modular cusp forms of weight k and level N . It has long been known that these objects satisfy many interesting congruence relations. One very powerful method for studying the congruences obeyed by modular forms modulo powers of a fixed prime p is to embed this space into the p-adic Banach space S k (Γ 1 (N ), r) of r-overconvergent p-adic cusp forms, defined as in [Kat73] using sections of ω ⊗k on certain affinoid subdomains of X 1 (N ) obtained by removing discs of radius p −r around the supersingular points; this space has been used to great effect by Coleman and others ([Col96,Col97]).It is known that there is a Hecke action on S k (Γ 1 (N ), r), as with the classical spaces, and these operators are continuous; and moreover, at least for 0 < r < p p+1 , the Atkin-Lehner operator U is compact. There is a rich spectral theory for 1

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“…Progress is being made, however. Recent work of Buzzard and Kilford [BK05] shows us that the 2-adic eigencurve indeed looks quite simple at the boundary of the weight space (3) : it is the disjoint union of infinitely many p-adic annuli. Using this, Buzzard and Calegari [BC] have shown that the 2-adic eigencurve is proper over the weight space.…”

Section: Generalisationsmentioning

confidence: 99%

Algebraic groups

2007

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The 2-adic eigencurve at the boundary of weight space

Abstract: We prove that, near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of 'evenly spaced' annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

Cited by 52 publications

References 5 publications

On the slopes of the {U_5} operator acting on overconvergent modular forms

On the slopes of the {U_5} operator acting on overconvergent modular forms

Spectral Expansions of Overconvergent Modular Functions

Algebraic groups

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