The eigencurve over the boundary of weight space

Liu, Ruochuan; Wan, Daqing; Xiao, Liang

doi:10.1215/00127094-0000012x

Cited by 35 publications

(57 citation statements)

References 24 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This proves (11) for M ≫ 0 as claimed. Likewise, to show (b) it is enough to prove that We note the following obvious facts we will use throughout.…”

Section: 3supporting

confidence: 69%

“…The persistence of this behavior ρ-by-ρ is neither completely surprising, nor does it follow from any of the partial results (e.g. [4,11]) on the actual spectral halos.…”

Section: Introductionmentioning

confidence: 84%

“…Write To show (a) we may show the stronger assertion that t < M +M 1/2+ε when M ≫ 0. Equivalently, by the choice of t, it is enough to show (11) u…”

Section: 3mentioning

confidence: 99%

See 2 more Smart Citations

Slopes of modular forms and the ghost conjecture, II

Bergdall

Pollack

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

In a previous article, we constructed an entire power series over p-adic weight space (the 'ghost series') and conjectured, in the Γ 0 (N )-regular case, that this series encodes the slopes of overconvergent modular forms of any p-adic weight. In this paper, we construct 'abstract ghost series' which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation that is locally reducible at p. Ample numerical evidence is given for this new conjecture. Further, we prove that the slopes computed by any abstract ghost series satisfy a distributional result at classical weights (consistent with conjectures of Gouvêa) while the slopes form unions of arithmetic progressions at all weights not in Zp. Abstract ghost seriesThe data needed for an abstract ghost series is two functions d, d new : Z −→ Z and an integer k 0 with 0 ≤ k 0 < δ. For notation, if n is an integer then set k n = k 0 + nδ and also define d p = 2d + d new . By the end of this section, we will make three assumptions (G), (LG), and (ND) on d and d new . (In Section 4 we introduce a fourth axiom (QL) which implies (G) and (LG).) We first assume: (G) lim n→∞ d(n) = ∞ and lim n→−∞ d(n) + d new (n) = −∞.

show abstract

“…This proves (11) for M ≫ 0 as claimed. Likewise, to show (b) it is enough to prove that We note the following obvious facts we will use throughout.…”

Section: 3supporting

confidence: 69%

“…The persistence of this behavior ρ-by-ρ is neither completely surprising, nor does it follow from any of the partial results (e.g. [4,11]) on the actual spectral halos.…”

Section: Introductionmentioning

confidence: 84%

See 1 more Smart Citation

Slopes of modular forms and the ghost conjecture, II

Bergdall

Pollack

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The eigenvarieties mentioned above are all rigid analytic spaces, so they parameterize forms that have coefficients in Q p -algebras. Recently, there has been interest in studying forms with coefficients in characteristic p. Liu, Wan, and Xiao [LWX17] constructed Z p Z × p -modules of automorphic forms for definite quaternion algebras. By taking quotients of this module, one can obtain both traditional p-adic automorphic forms and forms with coefficients in F p Z × p whose existence had been conjectured by Coleman.…”

mentioning

confidence: 99%

Equidimensional adic eigenvarieties for groups with discrete series

Gulotta

2019

Alg. Number Th.

View full text Add to dashboard Cite

We extend Urban's construction of eigenvarieties for reductive groups G such that G(R) has discrete series to include characteristic p points at the boundary of weight space. In order to perform this construction, we define a notion of "locally analytic" functions and distributions on a locally Qp-analytic manifold taking values in a complete Tate Zp-algebra in which p is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on p-adic Lie groups given by Johansson and Newton.

show abstract

“…In [21], Liu, Wan and Xiao gave a conjectural, but general, framework in which to view the Buzzard-Kilford calculation (see [26] also). Namely, those authors have conjectured that the slopes of the U p -operator acting on spaces of overconvergent p-adic cuspforms at p-adic weights "near the boundary of weight space" are finite unions of arithmetic progressions whose initial terms are the slopes in explicit classical weight two spaces.…”

mentioning

confidence: 99%