Research

I am interested broadly in arithmetic geometry, with a particular emphasis on problems inspired by isogeny-based cryptography. The security of a number of post-quantum cryptographic protocols relies on the assumed hardness of computing an isogeny between supersingular elliptic curves over fields of large characteristic. My research uses number theory to illuminate mathematical aspects of this problem, with a particular emphasis on understanding the graph theoretic structure of the underlying isogeny graphs.

I also have collaborative work on more classical arithmetic geometry, including the study of mirror symmetry for K3 surfaces, and minimal heights on elliptic curves. Most of these projects involve a significant computational or experimental aspect, and I am generally excited by projects that use current computing capabilities to derive insight into theory.

Publications & Preprints

  1. Bo Lau J, Morrison T, Orvis E, Scullard G, Zobernig L Zeta functions of abstract isogeny graphs and modular curves - arxiv.
  2. Davis R, Dukes J, Orvis E, Ribeiro T, Salerno A, Sturman L, Whitcher U Hypergeometric decomposition of Delsarte K3 pencils - arxiv.
  3. Cats S, Clark J, Curco-Iranzo M, Dombrowsky C, Maughn K, Orvis E Experimental investigations on Lehmer's Conjecture for elliptic curves - to appear LuCaNT 2025.
  4. Orvis, E Distribution of cycles in supersingular $\ell$-isogeny graphs, Journal of Number Theory, Volume 277, 2025, 236-261; arxiv.
  5. Jones, RF, Illig, P, Orvis, E, Segawa, Y, Spinale, N Newly Reducible Polynomial Iterates, Int. J. Number Theory 17(2021), no. 6, 1405-1427; arxiv.

Upcoming Conference Talks

  1. JMM, January 2026
    1. Special Session on Algebraic and Analytic aspects of L-functions
    2. Special Session on Mathematics of Isogeny-based crypgtography
    3. Special Session on Rethinking Number Theory: Highlighting the Research Directions of the RNT Workshop
  2. Front Range Algebra, Geometry, and Number Theory Seminar, Colorado State Univeristy, January 2026

Past Invited Talks

  1. Ihara zeta functions of abstract isogeny graphs and modular curves
    1. Isogeny Club, Online, (slides & recording), November 2025
    2. AMS Southeastern Sectional Meeting, Tulane University , October 2025
  2. Cycles in supersingular isogeny graphs: how to count and find them
    1. Swissogeny Day, ETH Zurich, June 2025
  3. On the distribution of cycles in supersingular $\ell$-isogeny graphs
    1. Leuven Isogeny Days, KU Leuven, September 2024
    2. AMS Eastern Sectional Meeting, Howard University, April 2024
Iterative roots