Mathematical Sciences: Arithmetic on Elliptic Curves and Related Topics

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This research will be on problems related to the question of supersingular primes for a given elliptic curve over a number field. Immediate goals include a proof of a natural sufficient condition for the infinitude of supersingular primes in the totally imaginary case and unconditional explicit lower bounds on the density of such primes. Further goals will include the nonzero fixed trace problem (of which many old classical problems are special cases) and various other open problems in the arithmetic of elliptic curves and abelian varieties of higher dimension. The area of this research is in that branch of number theory which studies certain cubic equations and the number of rational solutions of them (elliptic curves). The subject has connections throughout mathematics and has been instrumental in solving many problems in number theory concerning many topics. The number of solutions of the equation modulo primes is a central focus in the subject and this is the area focused on by this investigator.