Dr Dan Fretwell

Lecturer - Security and Protection Science

Research Interests

My main area of research is (Algebraic) Number Theory. In particular, I work on problems that involve a wide range of objects, from extremely explicit to mind bogglingly abstract:

  • Quadratic forms, Lattices, Codes, Integer valued polynomials, Diophantine equations.
  • Modular forms, Elliptic curves, Number Fields, Clifford algebras, Quaternion algebras.
  • Automorphic forms, Galois representations, L-functions, Bloch-Kato Selmer groups.
I like to work on explicit problems concerning the above (e.g. How do I compute this thing? How do I count these things? How do I tell whether two of these things are different?), but I also like much deeper problems (e.g. Does what I just did apply for more general families of automorphic form/Galois representations? How does what I just did fit into the general framework of the Langlands program? Have I just proved a special case of the Bloch-Kato conjecture?). Some of my work is motivated by deep conjectures in Mathematics, and the practical need to compute explicit evidence for these. I have very wide interests, extending beyond Number Theory. For example, I have recently been working on projects relating to Combinatorial Probability. In particular, I have studied a specific family of Markov chains called Interacting Particle Systems. These are useful in Statistical Mechanics and Physics (e.g. they can be used to model traffic flow). By comparing dynamics/probability measures on such systems one can often prove q-series identities of combinatorial significance (e.g. the famous Jacobi triple product identity). I am currently thinking about how my research can be useful in the real world (e.g. Cybersecurity, Cryptography and Quantum Computation). Have any ideas about this? Send me an email!

Selected Publications

Definite orthogonal modular forms: computations, excursions, and discoveries
Assaf, E., Fretwell, D., Ingalls, C., Logan, A., Secord, S., Voight, J. 31/12/2022 In: Research in Number Theory. 8, 4
Journal article

Interacting particle systems and Jacobi style identities
Balázs, M., Fretwell, D., Jay, J. 30/09/2022 In: Research in the Mathematical Sciences. 9, 3
Journal article

Hecke operators on Hilbert-Siegel theta series
Fretwell, D., Walling, L. 4/10/2021 In: International Journal of Number Theory. 17, 9, p. 1965-1996. 32 p.
Journal article

Automorphic forms for some even unimodular lattices
Dummigan, N., Fretwell, D. 30/04/2021 In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg . 91, 1, p. 29-67. 39 p.
Journal article

Smooth Values of Polynomials
Fretwell, D., Wooley, T., Bober, J., Martin, G. 1/04/2020 In: Journal of the Australian Mathematical Society. 108, 2, p. 245-261. 17 p.
Journal article

Genus 2 paramodular Eisenstein congruences
Fretwell, D. 14/06/2018 In: The Ramanujan Journal.
Journal article

Generic level p Eisenstein congruences for GSp4
Fretwell, D. 11/2017 In: Journal of Number Theory.
Journal article

Ramanujan-style congruences of local origin
Fretwell, D., Dummigan, N. 10/2014 In: Journal of Number Theory.
Journal article

Algebra and Geometry

B49, B - Floor, Fylde College
  • Algebra and Geometry