Ramanujan's famous congruence $\tau(p)\equiv 1+p^{11}\pmod{691}$ (for all primes $p$), where $\sum\tau(n)q^n:=q\prod(1-q^n)^{24}$, 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 $\zeta(12)/pi^{12}$, where $\zeta(s)=\sum 1/n^{s}$ 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 $\mathrm{GSp}_4$ 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 $\mathbb{Q}$. 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 $\lambda$ reducibility of Galois representations. From this, often it is possible to construct elements of order $\lambda$ 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.