Professor Neil Dummigan
School of Mathematical and Physical Sciences
Professor
+44 114 222 3713
Full contact details
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 Lfunction. (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 Lfunctions arising from modular forms. In accord with Langlands' vision, these Lfunctions can be viewed either as motivic Lfunctions, coming from arithmetic algebraic geometry, or as automorphic Lfunctions, coming from analysis and representation theory. (Examplemodularity of elliptic curves over Q. The Lfunction of the elliptic curve, encoding numbers of points modulo all different primes, is also the Lfunction coming from the qexpansion 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 BlochKato conjecture. This is the general conjecture on the behaviour of motivic Lfunctions at integer points (of which special cases are Dirichlet's class number formula and the Birch and SwinnertonDyer conjecture). Where predictions arising from the BlochKato conjecture cannot be proved, sometimes they can be supported by numerical experiments.
These congruences often seem to arise somehow from the intimate connection between Lfunctions and Eisenstein series, e.g. through the appearance of Lvalues in the constant terms of Eisenstein series, or when integrals are unfolded, e.g. in pullback formulas.
 Publications

Show: Featured publications All publications
Featured publications
Journal articles
 Automorphic Forms on Feit’s Hermitian Lattices. Experimental Mathematics. View this article in WRRO
 Eisenstein Congruences for SO(4, 3), SO(4, 4), Spinor, and Triple Product Lvalues. Experimental Mathematics, 27(2), 230250. View this article in WRRO
 Eisenstein congruences for split reductive groups. Selecta Mathematica (New Series), 22(3), 10731115. View this article in WRRO
 Quadratic Qcurves, units and Hecke Lvalues. Mathematische Zeitschrift, 280(34), 10151029. View this article in WRRO
All publications
Journal articles
 Congruences of local origin and automorphic induction. International Journal of Number Theory.
 GL2xGSp2 Lvalues and Hecke eigenvalue congruences. Journal de Theorie des Nombres de Bordeaux, 31(3), 751775. View this article in WRRO
 Automorphic Forms on Feit’s Hermitian Lattices. Experimental Mathematics. View this article in WRRO
 Kurokawa–Mizumoto congruences and degree8 Lvalues. Manuscripta Mathematica. View this article in WRRO
 Eisenstein Congruences for SO(4, 3), SO(4, 4), Spinor, and Triple Product Lvalues. Experimental Mathematics, 27(2), 230250. View this article in WRRO
 View this article in WRRO Lifting congruences to weight 3/2. Journal of the Ramanujan Mathematical Society, 32(4), 431440.
 Lifting puzzles and congruences of Ikeda and Ikeda–Miyawaki lifts. Journal of the Mathematical Society of Japan, 69(2), 801818. View this article in WRRO
 Eisenstein congruences for split reductive groups. Selecta Mathematica (New Series), 22(3), 10731115. View this article in WRRO
 Quadratic Qcurves, units and Hecke Lvalues. Mathematische Zeitschrift, 280(34), 10151029. View this article in WRRO
 Ramanujanstyle congruences of local origin. Journal of Number Theory, 143, 248261. View this article in WRRO
 Exact holomorphic differentials on a quotient of the Ree curve. Journal of Algebra, 400, 249272. View this article in WRRO
 A simple trace formula for algebraic modular forms. Experimental Mathematics.
 Powers of 2 in modular degrees of modular abelian varieties. Journal of Number Theory, 133(2), 501522.
 Yoshida lifts and Selmer groups. Journal of the Mathematical Society of Japan, 64, 13531405.
 Some Siegel modular standard Lvalues, and ShafarevichTate groups. Journal of Number Theory, 131(7), 12961330.
 Symmetric square Lvalues and dihedral congruences for cusp forms. Journal of Number Theory, 130(9), 20782091.
 Triple product Lvalues and dihedral congruences for cusp forms. International Mathematics Research Notices, 2010(10), 17921815.
 Critical values of symmetric power Lfunctions. Pure and Applied Mathematics Quarterly, 5(1), 127161.
 Symmetric square LFunctions and shafarevichtate groups, II. International Journal of Number Theory, 5(7), 13211345.
 Euler factors and local root numbers for symmetric powers of elliptic curves. Pure and Applied Mathematics Quarterly, 5(4), 13111341.
 Rational points of order 7. Bulletin of the London Mathematical Society, 40(6), 10911093.
 Eisenstein primes, critical values and global torsion. Pacific Journal of Mathematics, 233(2), 291308.
 On a conjecture of Watkins. Journal de Theorie des Nombres de Bordeaux, 18, 345355.
 Values of a Hilbert modular symmetric square Lfunction and the BlochKato conjecture. Journal of the Ramanujan Mathematical Society, 20(3), 167187.
 Rational torsion on optimal curves. International Journal of Number Theory, 1, 513531.
 Tamagawa factors for certain semistable representations. Bulletin of the London Mathematical Society, 37(6), 835845.
 Congruences of modular forms and tensor product Lfunctions. Bulletin of the London Mathematical Society, 36(2), 205215.
 Tamagawa factors for symmetric squares of TATE curves. MATHEMATICAL RESEARCH LETTERS, 10(56), 747762.
 Tamagawa factors for symmetric squares of Tate curves. Mathematical Research Letters, 10(56), 747762.
 Symmetric squares of elliptic curves: Rational points and selmer groups. Experimental Mathematics, 11(4), 457464.
 Symmetric square lfunctions and shafarevichtate groups. Experimental Mathematics, 10(3), 383400.
 Congruences of modular forms and Selmer groups. Mathematical Research Letters, 8(4), 479494.
 Period ratios of modular forms. Mathematische Annalen, 318(3), 621636.
 Complete pdescent for Jacobians of Hermitian curves. Compositio Mathematica, 119(2), 111132.
 Lower bounds for the minima of certain symplectic and unitary group lattices. American Journal of Mathematics, 121(4), 889918.
 Congruences for Certain Theta Series. Journal of Number Theory, 71(1), 86105.
 The representation of integers by binary additive forms. Compositio Mathematica, 111(1), 1533.
 Algebraic Cycles and Even Unimodular Lattices. Journal of the London Mathematical Society, 56(2), 209221.
 Symplectic Group Lattices as Mordell–Weil Sublattices. Journal of Number Theory, 61(2), 365387.
 The Determinants of Certain MordellWeil Lattices. American Journal of Mathematics, 117(6), 14091409.
 Quinary forms and paramodular forms. Mathematics of Computation.
 Lifting congruences to halfintegral weight. Research in Number Theory.
 Twisted adjoint Lvalues, dihedral congruence primes and the BlochKato conjecture. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg.
 Automorphic forms for some even unimodular lattices. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg.
 Congruences of SaitoKurokawa lifts and denominators of central spinor Lvalues. Glasgow Mathematical Journal.
Chapters
 Constructing elements in ShafarevichTate groups of modular motives In Reid M & Skorobogatov A (Ed.), Number Theory and Algebraic Geometry (pp. 91118). Cambridge University Press
Conference proceedings papers
 Research group
 Grants

Past grants, as Principal Investigator
Congruences of Siegel Modular Forms EPSRC
 Teaching activities

MAS211 Advanced Calculus and Linear Algebra MAS345 Codes and Cryptography