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

  • Dummigan N & Schönnenbeck S (2019) . Experimental Mathematics, Online first, 0-0.
  • Bergström J, Dummigan N, Mégarbané T, Ibukiyama T & Katsurada H (2018) . Experimental Mathematics, 27(2), 230-250.
  • Bergström J & Dummigan N (2016) . Selecta Mathematica (New Series), 22(3), 1073-1115.
  • Dummigan N & Golyshev V (2015) . Mathematische Zeitschrift, 280(3-4), 1015-1029.

All publications

Journal articles

  • Dummigan N, Pacetti A, Rama G & Tornaria G (2024) . Mathematics of Computation, 93, 1805-1858.
  • Dummigan N (2022) . Research in Number Theory, 8.
  • Dummigan N (2022) . Glasgow Mathematical Journal, 64(2), 504-525.
  • Dummigan N & Fretwell D (2021) . Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 91(1), 29-67.
  • Dummigan N & Spencer D (2021) . International Journal of Number Theory, 17(07), 1617-1629.
  • Dummigan N (2020) . Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 90(2), 215-227.
  • Bergström J, Dummigan N, Farmer D & Koutsoliotas S (2020) . Journal de Theorie des Nombres de Bordeaux, 31(3), 751-775.
  • Dummigan N & Schönnenbeck S (2019) . Experimental Mathematics, Online first, 0-0.
  • Dummigan N, Heim B & Rendina A (2019) . Manuscripta Mathematica, 160, 217-237.
  • Bergström J, Dummigan N, Mégarbané T, Ibukiyama T & Katsurada H (2018) . Experimental Mathematics, 27(2), 230-250.
  • Dummigan N & Krishnamoorthy S (2017) Lifting congruences to weight 3/2. Journal of the Ramanujan Mathematical Society, 32(4), 431-440.
  • Dummigan N (2017) . Journal of the Mathematical Society of Japan, 69(2), 801-818.
  • Bergström J & Dummigan N (2016) . Selecta Mathematica (New Series), 22(3), 1073-1115.
  • Dummigan N & Golyshev V (2015) . Mathematische Zeitschrift, 280(3-4), 1015-1029.
  • Dummigan N & Fretwell D (2014) . Journal of Number Theory, 143, 248-261.
  • Dummigan N & Farwa S (2014) . Journal of Algebra, 400, 249-272.
  • Dummigan NP (2013) . Experimental Mathematics.
  • Dummigan N & Krishnamoorthy S (2013) . Journal of Number Theory, 133(2), 501-522.
  • Dummigan NP, B"ocherer S & Schulze-Pillot R (2012) Yoshida lifts and Selmer groups. Journal of the Mathematical Society of Japan, 64, 1353-1405.
  • Dummigan N, Ibukiyama T & Katsurada H (2011) . Journal of Number Theory, 131(7), 1296-1330.
  • Dummigan N & Heim B (2010) . Journal of Number Theory, 130(9), 2078-2091.
  • Dummigan N & Heim B (2010) . International Mathematics Research Notices, 2010(10), 1792-1815.
  • Dummigan N & Watkins M (2009) . Pure and Applied Mathematics Quarterly, 5(1), 127-161.
  • Dummigan N (2009) . International Journal of Number Theory, 5(7), 1321-1345.
  • Dummigan N, Martin P & Watkins M (2009) . Pure and Applied Mathematics Quarterly, 5(4), 1311-1341.
  • Dummigan N (2008) . Bulletin of the London Mathematical Society, 40(6), 1091-1093.
  • Dummigan N (2007) . Pacific Journal of Mathematics, 233(2), 291-308.
  • Dummigan NP (2006) . Journal de Theorie des Nombres de Bordeaux, 18, 345-355.
  • Dummigan N (2005) . Bulletin of the London Mathematical Society, 37(6), 835-845.
  • Dummigan NP (2005) Values of a Hilbert modular symmetric square L-function and the Bloch-Kato conjecture. Journal of the Ramanujan Mathematical Society, 20(3), 167-187.
  • Dummigan NP (2005) . International Journal of Number Theory, 1, 513-531.
  • Dummigan N (2004) . Bulletin of the London Mathematical Society, 36(2), 205-215.
  • Dummigan N (2003) . MATHEMATICAL RESEARCH LETTERS, 10(5-6), 747-762.
  • Dummigan N (2003) . Mathematical Research Letters, 10(5-6), 747-762.
  • Dummigan N (2002) . Experimental Mathematics, 11(4), 457-464.
  • Dummigan N (2001) . Experimental Mathematics, 10(3), 383-400.
  • Dummigan N (2001) . Mathematical Research Letters, 8(4), 479-494.
  • Dummigan N (2000) . Mathematische Annalen, 318(3), 621-636.
  • Dummigan N (1999) . Compositio Mathematica, 119(2), 111-132.
  • Dummigan N & Tiep PH (1999) . American Journal of Mathematics, 121(4), 889-918.
  • Dummigan N & Tiep PH (1998) . Journal of Number Theory, 71(1), 86-105.
  • Bennett MA, Dummigan NP & Wooley TD (1998) . Compositio Mathematica, 111(1), 15-33.
  • Dummigan N (1997) . Journal of the London Mathematical Society, 56(2), 209-221.
  • Dummigan N (1996) . Journal of Number Theory, 61(2), 365-387.
  • Dummigan N (1995) . American Journal of Mathematics, 117(6), 1409-1409.

Book 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

Conference proceedings

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

Grants

Past grants, as Principal Investigator
EPSRC
Teaching activities
MAS211 Advanced Calculus and Linear Algebra
MAS345 Codes and Cryptography

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