Supersingular Distribution, Congruence Class Bias, and A Refinement of Strong Multiplicity One
<p>This thesis consists of four chapters, including an introduction.</p> <p>In Chapter 2, we take an averaging approach to the question of the distribution of supersingular primes of degree one, for elliptic curves over a number field. We begin by modifying the Lang-Trotter heur...
| Summary: | <p>This thesis consists of four chapters, including an introduction.</p>
<p>In Chapter 2, we take an averaging approach to the question of the distribution of supersingular primes of degree one, for elliptic curves over a number field. We begin by modifying the Lang-Trotter heuristic to address the case of an abelian extension, then we show that it holds on average (up to a constant) for a family of elliptic curves by using ideas of David-Pappalardi.</p>
<p>In Chapter 3, we prove constructively that there exists an infinite number of (arbitrarily) thin families of rational elliptic curves for which the Lang-Trotter conjecture holds on average, in part by using techniques of Fouvry-Murty.</p>
<p>In Chapter 4, we obtain a result related to the strong multiplicity one theorem for non-dihedral cuspidal automorphic representations for GL(2), with trivial central character and non-twist-equivalent symmetric squares. Given a real algebraic number, we also find a lower bound for the lower density of the set of finite places for which the associated Hecke eigenvalue is not equal to that algebraic number.</p> |
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