Ramanujan's famous congruence τ(p) ≡ 1+p
^{11} mod 691 (for all primes p), where ∑τ(n)q
^{n}:=q∏(1−q
^{n})
^{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)/pi
^{12}, where
ζ(s)=∑1/n
^{s} 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 GSp
_{4} 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. Also important is
the appearance of L-values in formulas for the norms of various theta-lifts.