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Families of modular forms. (English) Zbl 1052.11036
The author gives a down-to-earth introduction to the theory of $$p$$-adic families of modular forms, and presents an elementary proof of D. Wan’s result [Invent. Math. 133, No. 2, 449–463 (1998; Zbl 0907.11016)] that the Newton polygon of the $$U_p$$-operator acting on $$S_k(\Gamma_1(N_p))$$ is bounded below by an explicit quadratic lower bound which is independent of $$k$$ (Theorem 3).

MSC:
 11F85 $$p$$-adic theory, local fields 11F11 Holomorphic modular forms of integral weight
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References:
 [1] Coleman, R., p-adic Banach spaces and families of modular forms. Invent. Math.127 (1997), 417-479. · Zbl 0918.11026 [2] Coleman, R., Mazur, B., The eigencurve. In Galois representations in arithmetic algebraic geometry (Durham, 1996), CUP1998, 1-113. · Zbl 0932.11030 [3] Gouvêa, F., Mazur, B., Families of modular eigenforms. Math. Comp.58 no. 198 (1992), 793-805. · Zbl 0773.11030 [4] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Princeton University Press, 1994. · Zbl 0872.11023 [5] Taylor, R., Princeton PhD thesis. [6] Ulmer, D., Slopes of modular forms. Contemp. Math.174 (1994), 167-183. · Zbl 0853.11037 [7] Wan, D., Dimension variation of classical and p-adic modular forms. Invent. Math.133 (1998), 449-463. · Zbl 0907.11016
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