Dr Dan Fretwell FHEA

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, error correcting 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.
Some of my work is motivated by the Langlands program, which aims to explain deep links between these objects. One of my main focusses is the computation of explicit evidence towards these conjectures, and concrete applications of the general theory. I have very wide interests, extending beyond number theory. For example, I have recently been working with interacting particle systems. These are continuous-time Markov chains on discrete state spaces, having many applications in Statistical Mechanics and Physics (e.g. modelling traffic flow, data flow in networks and tumour growth). By comparing dynamics/probability measures coming from 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

Winstanley College
School Engagement

Rochdale Sixth Form College
School Engagement

  • Algebra and Geometry