Arithmetic partial differential equations, II: modular curves

Details Arithmetic partial differential equations, II: modular curves Arithmetic partial differential equations, II: modular curves

2008-04-30

arxiv.org/abs/0804.4856v1

Mathematics

Number Theory

Scientific paper

We classify ``arithmetic convection equations'' on modular curves, and describe 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 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 modular curves 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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