Abstract
Gouvêa and Mazur (Math Comp 58:793-805, 1992) made a conjecture on the local constancy of slopes of modular forms when the weight varies p-adically. Since one may decompose the space of modular forms according to associated residual Galois representations, the Gouvêa–Mazur conjecture makes sense for each such component. We prove the localized Gouvêa–Mazur conjecture when the residual Galois representation is irreducible and its restriction to \({{\,\textrm{Gal}\,}}\left( \bar{\mathbb {Q}}_p/\mathbb {Q}_p \right) \) is reducible and generic.
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The authors guarantee that the code used to generate the figure, which aids in estimating the minimal values, are available from the corresponding author, upon reasonable request.
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Acknowledgements
This manuscript would not have been conceivable without the insightful contributions derived from the works of Ruochuan Liu, Nha Truong, Liang Xiao, and Bin Zhao [6, 7]. We extend our profound gratitude to Liang Xiao in particular, for numerous enlightening discussions concerning the ghost conjecture. Furthermore, we wish to express our sincere appreciation to the anonymous referees for their invaluable recommendations on enhancing the structure and readability of our document. This work is partially supported by the Chinese NSF grant KRH1411532, two grants KBH1411247 and KBH1411268 from Shanghai Science and Technology Development Funds, and a grant from the New Cornerstone Science Foundation.
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Ren, R. The localized Gouvêa–Mazur conjecture. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02858-0
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DOI: https://doi.org/10.1007/s00208-024-02858-0