Daniel Vargas

Thesis Advisors: Dr. Dagan Karp (Harvey Mudd College - Department of Mathematics) and Dr. Yifeng Huang (University of Southern California - Department of Mathematics)

Second Reader: Dr. Ken Ono (University of Virginia - Department of Mathematics)

Contact: dvargas@g.hmc.edu

Galois theory and the arithmetic-geometric series

Arithmetic-geometric means have been used on the real numbers for centuries, such as by Euler to find a new way to compute π and by Gauss to evaluate elliptic integrals, but they have not been studied at all over over finite fields until recently, possibly due to difficulties in determining the “correct” geometric mean or possibly because a finite field has only finitely many elements, leading to inescapable loops. However, the eventual periodicity for finite fields actually makes them more interesting because finite fields have combinatorial structures that don’t exist elsewhere, providing us with new tools. Recent works have found that graphing the AGM over finite fields with order congruent to 3 modulo 4 yields a structure called a "jellyfish" (because of its shape), which charts elliptic curve isogeny networks, and find intriguing connections to class numbers and finite field hypergeometric functions. My thesis will address the open problem of extending this work to more fields, such as finite fields with order congruent to 5 modulo 8 and even to ℂ. This reveals deep connections to class numbers, elliptic curves, and more—with applications for areas such as elliptic curve cryptography.