Arithmetic partial differential equations, II |
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Authors: | Alexandru Buium Santiago R. Simanca |
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Affiliation: | Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, United States |
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Abstract: | We continue the study of arithmetic partial differential equations initiated in [7] by classifying “arithmetic convection equations” on modular curves, and by describing their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions (Mori, 1995 [13], Buium, 2003 [4]) of the same modular forms; in this sense, our arithmetic convection equations can be seen as “unifying” the two types of expansions. The theory can be generalized to one of “arithmetic heat equations” on modular curves, but we prove that they do not carry “arithmetic wave equations.” Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation. |
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Keywords: | Differential algebra Fermat quotients Modular forms |
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