Off-campus UMass Amherst users: To download campus access dissertations, please use the following link to log into our proxy server with your UMass Amherst user name and password.

Non-UMass Amherst users: Please talk to your librarian about requesting this dissertation through interlibrary loan.

Dissertations that have an embargo placed on them will not be available to anyone until the embargo expires.

#### Author ORCID Identifier

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Degree Program

Mathematics

#### Year Degree Awarded

2019

#### Month Degree Awarded

September

#### First Advisor

Thomas Weston

#### Subject Categories

Number Theory

#### Abstract

Let *E*_{1} x *E*_{2} over **Q** be a fixed product of two elliptic curves over **Q** with complex multiplication. I compute the probability that the *p*th Fourier coefficient of *E*_{1} x *E _{2}*, denoted as

*a*

_{p}(

*E*

_{1}) +

*a*(

_{p}*E*), is a square modulo

_{2}*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 *a _{p}* as primes

*p*and elliptic curves vary. The purpose is to test the conjecture that

*a*of an elliptic curve is a square modulo

_{p}*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

*a*are evenly distributed between squares and non-squares modulo

_{p}*p*asymptotically.

#### Recommended Citation

Nguyen, Vy Thi Khanh, "Elliptic Curves And Power Residues" (2019). *Doctoral Dissertations*. 1802.

https://scholarworks.umass.edu/dissertations_2/1802