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Divisors Of Modular Parameterizations Of Elliptic Curves, Jonathan Reid Hales
Divisors Of Modular Parameterizations Of Elliptic Curves, Jonathan Reid Hales
Theses and Dissertations
The modularity theorem implies that for every elliptic curve E /Q there exist rational maps from the modular curve X_0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y(z) that satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be constructed to calculate the q - expansions of these parameterizations at any cusp. These functions are algebraic over Q(j(z)) and satisfy modular polynomials where each of the coefficient functions are rational functions in …
Congruences For Coefficients Of Modular Functions In Levels 3, 5, And 7 With Poles At 0, Ryan Austin Keck
Congruences For Coefficients Of Modular Functions In Levels 3, 5, And 7 With Poles At 0, Ryan Austin Keck
Theses and Dissertations
We give congruences modulo powers of p in {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and Jenkins and continuing work done by the Jenkins, the author, and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at infinity.