# 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.

## Career Details

I received my PhD in Mathematics from the University of Sheffield in 2015, under the supervision of Prof. Neil Dummigan. After, I was a Heilbronn Research Fellow at the University of Bristol until 2022, working on research of both theoretical and practical importance in Mathematics. I was a Lecturer of Mathematics at the University of South Wales 2022-2023.

## Web Links

## PhD Supervision Interests

Projects in Algebraic Number Theory and related areas (in particular projects concerning the arithmetic of Modular Forms and Galois Representations). Interested in projects concerning applications to Cryptography and Cyber Security.

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

All Publications

Algebra and Geometry

- Algebra and Geometry