Slopes of Modular Forms

@article{Buzzard2016SlopesOM,
  title={Slopes of Modular Forms},
  author={Kevin Buzzard and Toby Gee},
  journal={arXiv: Number Theory},
  year={2016},
  pages={93-109}
}
We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in Buzzard (Asterisque 298:1–15, 2005), discuss strategies for making further progress, and examine other related questions. 

Paper Mentions

Blog Post
Slopes of modular forms and the ghost conjecture (unabridged version)
We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes andExpand
Slopes of modular forms and the ghost conjecture, II
We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes andExpand
NOTES ON THE GHOST CONJECTURE: A QUALITATIVE APPROACH TO PREDICTING p-ADIC SLOPES OF MODULAR FORMS
These notes were presented at a joint workshop between Boston University and Keio University in September 2015. The work presented is still in progress, jointly with Rob Pollack of Boston University.Expand
A remark on non-integral p -adic slopes for modular forms
Abstract We give a sufficient condition, namely “Buzzard irregularity”, for there to exist a cuspidal eigenform which does not have integral p -adic slope.
OVERCONVERGENT MODULAR FORMS
In these notes, we aim to give a friendly introduction to the theory of overconvergent modular forms and some examples of recent arithmetic applications. The emphasis is on explicit examples andExpand
Reductions of Galois representations for slopes in $(1,2)$
We describe the semisimplifications of the mod p reductions of certain crystalline two-demensional local Galois representations of slopes in (1, 2) and all weights.The proof uses the compatibilityExpand
Reduction modulo p of two-dimensional crystalline representations of G_{Q_p} of slope less than three
We use the p-adic local Langlands correspondence for GL_2(Q_p) to find the reduction modulo p of certain two-dimensional crystalline Galois representations. In particular, we resolve a conjecture ofExpand
Upper bounds for constant slope p-adic families of modular forms
We study $p$-adic families of eigenforms for which the $p$-th Hecke eigenvalue $a_p$ has constant $p$-adic valuation ("constant slope families"). We prove two separate upper bounds for the size ofExpand
Slopes of Overconvergent Hilbert Modular Forms
TLDR
An explicit description of the matrix associated to the Up operator acting on spaces of overconvergent Hilbert modular forms over totally real fields is given and a lower bound on the Newton polygon of the Up is proved. Expand
The Jacquet–Langlands correspondence for overconvergent Hilbert modular forms
  • C. Birkbeck
  • Mathematics
  • International Journal of Number Theory
  • 2019
We use results by Chenevier to interpolate the classical Jacquet–Langlands correspondence for Hilbert modular forms, which gives us an extension of Chenevier’s results to totally real fields. FromExpand
...
1
2
...

References

SHOWING 1-10 OF 52 REFERENCES
Construction of some families of 2-dimensional crystalline representations
Abstract.We construct explicitly some analytic families of étale (φ,Γ)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of ourExpand
Slopes of overconvergent 2-adic modular forms
We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation.Expand
Explicit reduction modulo p of certain 2-dimensional crystalline representations, II
We complete the calculations begun in [BG09], using the p-adic local Langlands correspondence for GL2(Q_p) to give a complete description of the reduction modulo p of the 2-dimensional crystallineExpand
A counterexample to the Gouvêa–Mazur conjecture
Abstract Gouvea and Mazur made a precise conjecture about slopes of modular forms. Weaker versions of this conjecture were established by Coleman and Wan. In this Note, we exhibit examplesExpand
A family of Calabi-Yau varieties and potential automorphy
We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j-invariant definedExpand
Patching and the p-adic local Langlands correspondence
We use the patching method of Taylor--Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL_n(F), F a finite extension of Q_p. We use our construction to proveExpand
Explicit Reduction Modulo p of Certain Two-Dimensional Crystalline Representations
In this paper, we use the p-adic local Langlands correspondence for to explicitly compute the reduction modulo p of certain two-dimensional crystalline representations of small slope, and giveExpand
Dimension variation of classical and p-adic modular forms
Abstract. A quadratic bound is obtained for a conjecture of Gouvêa-Mazur on arithmetic variation of dimensions of classical and p-adic modular forms.
Modular forms and p-adic Hodge theory
For a modular form, Deligne constructs an associated `-adic representation of the Galois group GQ ˆ Gal… Q=Q†. We show that it is compatible with the local Langlands correspondence at p ˆ ` in theExpand
The 3-adic eigencurve at the boundary of weight space
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 aExpand
...
1
2
3
4
5
...