Isogeny graphs, zero-cycles, and modular forms : computations over algebraic curves and surfaces

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Author/Creator: Love, Jonathan Richard, author.
Contributor: Vakil, Ravi, degree supervisor. Thesis advisor; Venkatesh, Akshay, 1981- degree supervisor. Thesis advisor; Boneh, Dan, 1969- degree committee member. Thesis advisor; Stanford University. Department of Mathematics.

Summary: In the study of algebraic varieties over number fields or finite fields, there are many properties of these varieties that can be difficult to compute. This thesis discusses variants of the following three computational problems: (a) to compute an isogeny between two given supersingular elliptic curves; (b) to determine whether a zero-cycle on a product of elliptic curves is rationally equivalent to zero, and if so, to compute a rational equivalence; (c) to compute the space of cuspidal modular forms over a function field of genus greater than one. For each of these problems, we present algorithms that allow the problem to be solved in certain special cases, and prove results about these cases