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Generalized Atkin Polynomials and Non-Ordinary Hyperelliptic Curves

  • Author(s): Lane, Matthew Ernest;
  • Advisor(s): Duke, William;
  • et al.

There exists a sequence of orthogonal polynomials with many interesting properties from the standpoint of number theory. These polynomials are called Atkin polynomials, and they can be constructed using the theory of modular forms for the group PSL_{2}(Z). Closed formulas for these polynomials are known, and their zeros provide information about supersingular elliptic curves.

In this dissertation, we construct an infinite collection of sets of orthogonal polynomials, of which the Atkin polynomials are but one example. These polynomials are constructed using the theory of modular forms for the Hecke triangle groups G_{m}, as well as the theory of hypergeometric functions. As in the previously known case, the zeros of this larger family of polynomials provide information about curves. In this setting, however, the curves are hyperelliptic, and the zeros detect whether or not certain curves are ordinary. We show how these curves arise and give proofs generalizing the known properties of the Atkin polynomials. We also interpret these generalized Atkin polynomials within the framework of period functions and weakly holomorphic modular forms, and prove some new results on the Fourier coefficients of certain modular integrals.

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