Slopes of modular forms and the ghost conjecture, II
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- by John Bergdall and Robert Pollack PDF
- Trans. Amer. Math. Soc. 372 (2019), 357-388 Request permission
Abstract:
In a previous article we constructed an entire power series over $p$-adic weight space (the ghost series) and conjectured, in the $\Gamma _0(N)$-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 $\mathbf {Z}_p$.References
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Additional Information
- John Bergdall
- Affiliation: Department of Mathematics, Bryn Mawr College, Bryn Mawr, Pennsylvania 19010
- MR Author ID: 1039069
- Email: jbergdall@brynmawr.edu
- Robert Pollack
- Affiliation: Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, Massachusetts 02215
- MR Author ID: 715629
- Email: rpollack@math.bu.edu
- Received by editor(s): October 29, 2017
- Received by editor(s) in revised form: February 27, 2018
- Published electronically: November 5, 2018
- Additional Notes: The first author was supported by NFS grant DMS-1402005.
The second author was supported by NFS grants DMS-1303302 and DMS-1702178 as well as a fellowship from the Simons Foundation. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 357-388
- MSC (2010): Primary 11F33; Secondary 11F85
- DOI: https://doi.org/10.1090/tran/7549
- MathSciNet review: 3968772