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Author ORCID Identifier
Open Access Dissertation
Doctor of Philosophy (PhD)
Year Degree Awarded
Month Degree Awarded
Let E1 x E2 over Q be a fixed product of two elliptic curves over Q with complex multiplication. I compute the probability that the pth Fourier coefficient of E1 x E2, denoted as ap(E1) + ap(E2), is a square modulo p. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical assumption on the twists in various cases.
Next, I consider the sum of quadratic residue of ap as primes p and elliptic curves vary. The purpose is to test the conjecture that ap of an elliptic curve is a square modulo p about half of the time across prime numbers so that the sum is expected to be 0. Although the sum turns out to be positively biased, I show, assuming a natural independence result, that the ap are evenly distributed between squares and non-squares modulo p asymptotically.
Nguyen, Vy Thi Khanh, "Elliptic Curves And Power Residues" (2019). Doctoral Dissertations. 1802.