The 3-adic eigencurve at the boundary of weight space

Roe, David

doi:10.1142/s1793042114500560

Cited by 26 publications

(52 citation statements)

References 8 publications

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“…When the tame level is trivial and p = 2, this conjecture was verified using an explicit computation by Buzzard and Kilford [BK05], extending the thesis work of M. Emerton [Em98]. More explicit computations for small p and small tame levels have appeared in [Ja04,Kil08,KM12,Ro14]. A partial result that is independent of the prime p and the tame level was proved by J. Zhang and the second and third authors [WXZ14 + ].…”

Section: Introductionmentioning

confidence: 77%

The eigencurve over the boundary of weight space

Liu¹,

Wan²,

Xiao³

2017

Duke Math. J.

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We prove that the eigencurve associated to a definite quaternion algebra over $\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the $U_p$-slopes of points on each fixed connected component are proportional to the $p$-adic valuations of the parameter on weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves towards the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.Comment: 36 pages. Final version, to appear in Duke Math Journa

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Section: Introductionmentioning

confidence: 77%

The eigencurve over the boundary of weight space

Liu¹,

Wan²,

Xiao³

2017

Duke Math. J.

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“…This then implies that over the boundary of weight space the eigencurve looks like a countable union of annuli. For p = 2, 3 and trivial tame level this was proven by Buzzard-Kilford and Roe in [BK05, Therem B], [Roe14,Theorem 1]. For more details on the precise conjectures and their implications, see [BG16].…”

Section: Introductionmentioning

confidence: 80%

Slopes of Overconvergent Hilbert Modular Forms

Birkbeck

2019

Experimental Mathematics

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We give an explicit description of the matrix associated to the Up operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the centre and near the boundary of weight space for certain real quadratic fields. Near the boundary of weight space we see that the slopes do not appear to be given by finite unions of arithmetic progressions but instead can be produced by a simple recipe from which we make a conjecture on the structure of slopes. We also prove a lower bound on the Newton polygon of the Up.

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“…For higher slopes we get, using Proposition 3.11, that Theorem 3.10 then predicts that the slopes in S † z 2 χ (1) (scaled by 10) are given by the 55 arithmetic progressions with common difference 50 and starting terms: 0, 2, 3,4,5,5,6,7,8,10,10,11,12,13,14,14,15,16,17,19,19,20,21,22,23,23,24,25,27,28,28,29,30,31,32,32,33,35,36,37,37,38,39,40,41,41,43,44,45,46,46,47,48,49,50. Note that we've included in this list the contribution of the progression 10, 20, 30, .…”

Section: 12mentioning

confidence: 91%

Arithmetic properties of Fredholm series forp-adic modular forms

Bergdall

Pollack

2016

Proc. London Math. Soc.

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Abstract. We study the relationship between recent conjectures on slopes of overconvergent padic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the Up operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.

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The 3-adic eigencurve at the boundary of weight space

Abstract: This paper generalizes work of Buzzard and Kilford to the case p = 3, giving an explicit bound for the overconvergence of the quotient Eκ/V (Eκ) and using this bound to prove that the eigencurve is a union of countably many annuli over the boundary of weight space.

Cited by 26 publications

References 8 publications

The eigencurve over the boundary of weight space

The eigencurve over the boundary of weight space

Slopes of Overconvergent Hilbert Modular Forms

Arithmetic properties of Fredholm series forp-adic modular forms

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