2008/04/30 by Buium, Alexandru, Simanca, Santiago R.
#11F11 #11F32 #12H05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.0804.4856
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.