Lecturer of Mathematics,
University of Lancaster
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Honorary Research Fellow,
University of Bristol
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d (dot) fretwell (at) lancaster (dot) ac (dot) uk
daniel (dot) fretwell (at) bristol (dot) ac (dot) uk
Dr. Dan Fretwell
My PhD thesis, "Level p paramodular congruences of Harder type".
Publications:
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(with J. Bober, G. Kopp, L. Du and T. Wooley) On 2superirreducible polynomials over finite fields. Arxiv. To appear in Indagationes Mathematicae.

(with J. Roberts) Symmetric bilinear forms, superalgebras and integer matrix factorization. Linear Algebra and its Applications, Vol 700, p.6179 (2024). Arxiv.

(with J. Roberts) Newform Eisenstein congruences of local origin. Ramanujan Journal, Vol 64, Issue 2, p.505–527 (2024). Arxiv.

(with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight) Definite orthogonal modular forms: Computations, Excursions and Discoveries. Proceedings of ANTS XV, Research in Number Theory, Vol 8, Issue 4 (2022). Arxiv.

(with M. Balazs and J. Jay) Interacting particle systems and Jacobi style identities. Research in the Mathematical Sciences, Vol 9, Issue 3 (2022). Arxiv.

(with N. Dummigan) Automorphic forms for some even unimodular lattices. Abh. Math. Sem. Univ. Hamburg, Vol 91, Issue 1, p.2967 (2021). Supplementary data, Arxiv.

(with L. Walling) Hecke operators on HilbertSiegel theta series. International Journal of Number Theory, Vol 17, Issue 9, p.19651996 (2021). Arxiv.

(with J. Bober, G. Martin and T. Wooley) Smooth values of polynomials. Journal of the Australian Maths Society, Vol 108, Issue 2, p.245261 (2020). Arxiv.

Generic level p Eisenstein congruences for GSp4. Journal of Number Theory, Vol 180, p.673693 (2017). Arxiv.

Genus 2 paramodular Eisenstein Congruences. Ramanujan Journal, Vol 46, Issue 2, p.447473 (2017). Arxiv.

(with N. Dummigan) Ramanujanstyle congruences of local origin. Journal of Number Theory, Vol 143, p.248261 (2014).
In preparation:

(with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight) Orthogonal Modular Forms attached to Quaternary Lattices.

(with M. Balázs and J. Jay) Interacting Particle Systems, Lattice Paths, Generalised Frobenius Partitions and Linear Codes.

(with C. Hsu and D. Spencer) Hilbert modular Eisenstein congruences.

(with N. Gillespie and B. Naskrecki) Equiangular lines and an elliptic surface.
Past/present PhD students:

Jenny Roberts (University of Bristol, 2021present)
A selection of research talks:
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Interacting Particle Systems and Jacobi style identities, given at Purdue PANTHA seminar (2021).

(Real Quadratic) Arthurian Tales, given at Young Researchers in Algebraic Number Theory, Warwick (2019). Longer version given at International Automorphic Forms Seminar (2020).

Hilbert modular Eisenstein congruences, given at Young Researchers in Algebraic Number Theory, Sheffield (2018).

An Eisenstein congruence for genus 2 HilbertSiegel forms, given at the AMS Special Session on Real analytic automorphic forms, UNT, Texas (2017).

Paramodular Eisenstein congruences for GSp4, given at the 31st automorphic forms workshop, ETSU, Tennessee (2017).

Local origin congruences for elliptic modular forms, given at the 30th automorphic forms workshop, Wake Forest, North Carolina (2016).

Level p paramodular congruences of Harder type, given at the EU/US Building bridges automorphic forms workshop, Bristol (2014).

Towards a "convoluted" problem of Cohn, given at BMC, Bristol (2016).

A short 2min talk on local origin congruences, given at a workshop on Siegel and Bianchi modular forms, Sheffield (2014).
A selection of outreach/student talks:

Password Hacking, the De Bruijn way, given at the Matrix undergrad colloquium in Bristol (2019).

A oCmedy of Errosr, public outreach talk given at Pint of Science, Bristol (2018).

Escher and the Droste effect, given at the Matrix undergrad colloquium in Bristol (2017).

The magic of modular forms, given to Cambridge Part III students (2014).

The Eulerian quandry, given to Year 9 pupils at St. Bernard's School in Rotherham (2014).

Misc postgrad level talks, given at various seminars; The 290 theorem, Elliptic curves and the SatoTate conjecture, The Leech lattice and two remarkable properties, An instance of the Cebotarev density theorem.

Misc undergrad level talks, given at various undergrad colloquia; Primes of the form x^2+ny^2, When is a prime not prime?, Cyclotomic number fields.