Arithmetic properties of Fredholm series for p -adic modular forms

Trans. Amer. Math. Soc.

Pollack

2018

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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) = −∞.

“…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: 83%

Slopes of modular forms and the ghost conjecture, II

Trans. Amer. Math. Soc.

Pollack

2018

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“…Then, we were pleasantly astonished to observe that exactly m i (k)-many zeros of a i (w) were visibly close to w k (for instance, within 2 −9 and sometimes as close as 2 −36 or 2 −81 ). 3 The data from these computations can be found at the website [2].…”

Section: Statement Of Conjecturementioning

confidence: 99%

“…We view such agreement as compelling evidence for both conjectures. 3 We learned of this phenomenon, in the case of a1(w) and the weight k = 14, from an unpublished note of Buzzard.…”

Section: Comparison With Known or Conjectured Lists Of Slopesmentioning

confidence: 99%

“…Namely, one further expects that the list of slopes of NP(P (ε) ) is a finite union of arithmetic progressions (component-by-component). It is not included in the above statement since it is known to be implied by Conjecture 4.1 (see [3,Theorem B] and the proofs in [21]). In particular, the ghost conjecture implies the slopes of the mod p ghost series are a finite union of arithmetic progressions as well.…”

Section: Halos and Arithmetic Progressionsmentioning

confidence: 99%

“…In[3], the authors showed that if N = 1 then the coefficients a (ε) i (w) are not divisible by p. For N > 1 this is not true, but we don't believe this divisibility plays a crucial role for predicting slopes.…”

mentioning

confidence: 91%

See 2 more Smart Citations

Slopes of Modular Forms and the Ghost Conjecture

International Mathematics Research Notices

Pollack

2017

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Upper bounds for constant slope p-adic families of modular forms

2019

Sel. Math. New Ser.

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We study p-adic families of eigenforms for which the pth Hecke eigenvalue ap has constant p-adic valuation ("constant slope families"). We prove two separate upper bounds for the size of such families. The first is in terms of the logarithmic derivative of ap while the second depends only on the slope of the family. We also investigate the numerical relationship between our results and the former Gouvêa-Mazur conjecture.