Tristan Phillips

Hello! I am an NSF postdoc researching number theory at Dartmouth College. My postdoc mentors are John Voight and Asher Auel. I received my PhD from the University of Arizona, where my adviser was Bryden Cais (and where my unofficial co-adviser was Doug Ulmer).  I received my undergraduate degree in mathematics from Shippensburg University, where my adviser was Lenny Jones. 

 Here is my CV.

Paper(s) in Preparation:

Unbounded Average Selmer Ranks of Elliptic Curves in Torsion Families

Points of Bounded Height on Some Genus Zero Drinfeld Modular Curves

Preprint(s):

Counting Points on Some Genus Zero Shimura Curves (with Tyler Genao, Freddy Saia, Tim Santens, and John Yin) (arxiv)

100% of Elliptic Curves With a Marked Point Have Positive Rank (with Jun-Yong Park) (arxiv)

Points of Bounded Height in Images of Morphisms of Weighted Projective Stacks with Applications to Counting Elliptic Curves (arxiv)

Local Monodromy of 1-Dimensional p-Divisible Groups (arxiv)

Research Publications:

10. Average Analytic Ranks of Elliptic Curves over Number Fields, Forum Math Sigma 13 (2025), 36 pp. (arxiv)

9. Points of Bounded Height on Weighted Projective Spaces over Global Function Fields, Ramanujan Journal 65 (2024), 477-486. (arxiv)

8. Points of Bounded Height on Projective Spaces over Global Function Fields via Geometry of Numbers, Finite Fields and their Applications 96 (2024), 9 pp. (arxiv)

7. Counting Drinfeld Modules with Prescribed Local Conditions, Research in Number Theory 10 (2024) 22 pp. (arxiv)

6. Most Elliptic Curves Over Global Function Fields Are Torsion Free, Acta Arithmetica 202 (2022), 21-28. (arxiv)

5. Infinite Families of Monogenic Trinomials and Their Galois Groups (with Lenny Jones), International Journal of Mathematics 29 (2018), 11 pp.

4. An Infinite Family of Ninth Degree Dihedral Polynomials (with Lenny Jones), Bulletin of the Australian Mathematical Society 97 (2018), 47-53.

3. Arithmetic Progressions of Polygonal Numbers with Common Difference a Polygonal Number (with Lenny Jones), Integers 17 (2017), 8 pp.

2.  Primefree Shifted Lucas Sequences of the Second Kind (with Dan Ismailescu and Lenny Jones), Journal of Number Theory 173 (2017), 87-99.

1.  New Primitive Covering Numbers and Their Properties (with Joshua Harrington and Lenny Jones), Journal of Number Theory 172 (2017), 160-177.

Notes:

Notes on the Birch and Swinnerton-Dyer Conjecture. These were for a talk at an RTG workshop at The Ohio State in 2025. 

Notes motivating étale cohomology. These were for a learning seminar on arithmetic geometry at Dartmouth College in 2025. 

Notes on the Néron–Ogg–Shafarevich Criterion. These were for a learning seminar on arithmetic geometry at Dartmouth College in 2025. 

Department of Mathematics

Dartmouth College

Email: tristanphillips72(at)gmail(dot)com