Slopes of Overconvergent Hilbert Modular Forms

Abstract

We give an explicit description of the matrix associated to the U_p operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the center 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 U_p.

AMS Subject Classification:

• 11F41
• 11F33
• 11Y40

Acknowledgments

The author would like to thank his supervisor Lassina Dembélé for his support and guidance. He would also like to thank Fabrizio Andreatta, David Hansen, Alan Lauder and Vincent Pilloni for interesting discussions and very useful suggestions. Lastly, this work is part of the authors thesis so I wish to thank my examiners Kevin Buzzard and David Loeffler, as well as the referee for their very useful comments and corrections.

Funding

This study was supported by Engineering and Physical Sciences Research Council (EP/N509577/1).

Table 1. Split case, p = 3, classical slopes.

Table 2. Split case, p = 3, overconvergent slopes.

Table 7. Centre slopes for p = 2.

Notes

1 In fact one can show that they lie in ${\text{val}}_p(w(\kappa ))\le \frac{1}{p-1}$ cf. [Bergdall and Pollack 16, Lemma 1.6].

2 For p = 2 the centre is where ${\text{val}}_2(-)\ge 3$ and the boundary where ${\text{val}}_2(-)<3.$

3 Here the notation is such that if S is a sequence of slopes and $i\in \mathbb{Z}$, then we let S + i denote the set, where we add i to each slope in S.

4 This was shown to follow from (a) by Bergdall–Pollack in [Bergdall and Pollack 16, Theorem B].

5 For example, in some of our computations, we would need our approximation matrix to have $R\sim {10}^6$, although computations suggest that, in this case, we only need $R\sim 480$, but we cannot at this time prove this much stronger bound.

6 In fact $d=2,5,13,17$ are the only such examples, see [Kirschmer and Voight 10].

7 Note that here, for consistency, we are defining weight space over ${\mathcal{O}}_L$, but with more care one can work over ${\mathbb{Z}}_p$ which is more customary when discussing integral models, see [Andreatta et al. 16a, Section 2], but we do not need this here.

8 See [Buzzard and Calegari 05, Jacobs 03, Wan et al. 17] for other places where such functions are used.

9 Recall that we are using Notation 2.1.10.

10 Here h is the class number of (D, U).

11 For more details on Hodge polygons see [Wan et al. 17, Section 4.7] and [Kedlaya 10, Section 4.3].

12 This is most likely not the optimal bound.

13 This means we choose an ordering such that the first $\dim (S_{\kappa }(U))$ basis elements form a basis of $S_{\kappa }(U)$ (note that this ordering may differ from the one given by Bi as chosen in 4.1.24).

14 Specifically, we computed the slopes for many weights and they were always the same.

15 The appearance of $v_p(k)$ is due to our normalization of U_p.

Additional information

Funding

This study was supported by Engineering and Physical Sciences Research Council (EP/N509577/1)