Professor Neil Dummigan

School of Mathematical and Physical Sciences

Professor of Mathematics

n.p.dummigan@sheffield.ac.uk
+44 114 222 3713
J8, Hicks Building

Full contact details

Professor Neil Dummigan
School of Mathematical and Physical Sciences
J8
Hicks Building
Hounsfield Road
Sheffield
S3 7RH
Research interests

Ramanujan's famous congruence τ(p)≡1+p11(mod691) (for all primes p), where ∑τ(n)qn:=q∏(1−qn)24, is an example of a congruence involving the Hecke eigenvalues of a modular form, with a modulus coming from the algebraic part of a critical value of an L-function. (In this case, the prime 691 divides ζ(12)/pi12, where ζ(s)=∑1/ns is the Riemann zeta function.) I am interested in congruences involving the Hecke eigenvalues of modular forms, and more generally of automorphic representations for groups such as GSp4 and U(2,2), modulo primes appearing in critical values of various L-functions arising from modular forms. In accord with Langlands' vision, these L-functions can be viewed either as motivic L-functions, coming from arithmetic algebraic geometry, or as automorphic L-functions, coming from analysis and representation theory. (Example-modularity of elliptic curves over Q. The L-function of the elliptic curve, encoding numbers of points modulo all different primes, is also the L-function coming from the q-expansion of some modular form of weight 2.)

On the motivic side, there ought to be Galois representations associated to suitable automorphic representations, and in some cases this is known. Interpreting Hecke eigenvalues as traces of Frobenius elements, the congruences express the mod λ reducibility of Galois representations. From this, often it is possible to construct elements of order λ in generalised global torsion groups or Selmer groups, thereby proving consequences of the Bloch-Kato conjecture. This is the general conjecture on the behaviour of motivic L-functions at integer points (of which special cases are Dirichlet's class number formula and the Birch and Swinnerton-Dyer conjecture). Where predictions arising from the Bloch-Kato conjecture cannot be proved, sometimes they can be supported by numerical experiments.

These congruences often seem to arise somehow from the intimate connection between L-functions and Eisenstein series, e.g. through the appearance of L-values in the constant terms of Eisenstein series, or when integrals are unfolded, e.g. in pullback formulas.

Publications

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Journal articles

All publications

Journal articles

Chapters

  • Dummigan N, Stein W & Watkins M (2003) Constructing elements in Shafarevich-Tate groups of modular motives In Reid M & Skorobogatov A (Ed.), Number Theory and Algebraic Geometry (pp. 91-118). Cambridge University Press RIS download Bibtex download

Conference proceedings papers

  • Dummigan N (1999) Theta series congruences. Integral Quadratic Forms and Lattices (pp 249-252) RIS download Bibtex download
Research group

Number Theory

Grants

Past grants, as Principal Investigator
Congruences of Siegel Modular Forms EPSRC
Teaching activities
MAS211 Advanced Calculus and Linear Algebra
MAS345 Codes and Cryptography

Links

Professor Neil Dummigan's homepage