It is called a congruence ideal because when B is a
Hecke algebra and f is a homomorphism corresponding to a
modular form, the congruence ideal describes congruences between the modular form of f and other modular forms.
Example
Suppose C and D are rings with homomorphisms to a ring E, and let B = C×ED be the pullback, given by the subring of C×D of pairs (c,d) where c and d have the same image in E. If f is the natural projection from B to C, then the kernel is the ideal J of elements (0,d) where d has image 0 in E. If J has annihilator 0 in D, then its annihilator in B is just the kernel I of the map from C to E. So the congruence ideal of f is the ideal (I,0) of B.
Suppose that B is the
Hecke algebra generated by
Hecke operatorsTn acting on the 2-dimensional space of modular forms of level 1 and weight 12.This space is 2 dimensional, spanned by the Eigenforms given by the
Eisenstein seriesE12 and the
modular discriminant Δ. The map taking a Hecke operator Tn to its eigenvalues (σ11(n),τ(n)) gives a homomorphism from B into the ring Z×Z (where τ is the
Ramanujan tau function and σ11(n) is the sum of the 11th powers of the divisors of n). The image is the set of pairs (c,d) with c and d congruent mod 619 because of Ramanujan's congruence σ11(n) ≡ τ(n) mod 691. If f is the homomorphism taking (c,d) to c in Z, then the congruence ideal is (691). So the congruence ideal describes the congruences between the forms E12 and Δ.
References
Lenstra, H. W. (1995), "Complete intersections and Gorenstein rings", in
Coates, John (ed.), Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, pp. 99–109,
ISBN1-57146-026-8,
MR1363497,
Zbl0860.13012
It is called a congruence ideal because when B is a
Hecke algebra and f is a homomorphism corresponding to a
modular form, the congruence ideal describes congruences between the modular form of f and other modular forms.
Example
Suppose C and D are rings with homomorphisms to a ring E, and let B = C×ED be the pullback, given by the subring of C×D of pairs (c,d) where c and d have the same image in E. If f is the natural projection from B to C, then the kernel is the ideal J of elements (0,d) where d has image 0 in E. If J has annihilator 0 in D, then its annihilator in B is just the kernel I of the map from C to E. So the congruence ideal of f is the ideal (I,0) of B.
Suppose that B is the
Hecke algebra generated by
Hecke operatorsTn acting on the 2-dimensional space of modular forms of level 1 and weight 12.This space is 2 dimensional, spanned by the Eigenforms given by the
Eisenstein seriesE12 and the
modular discriminant Δ. The map taking a Hecke operator Tn to its eigenvalues (σ11(n),τ(n)) gives a homomorphism from B into the ring Z×Z (where τ is the
Ramanujan tau function and σ11(n) is the sum of the 11th powers of the divisors of n). The image is the set of pairs (c,d) with c and d congruent mod 619 because of Ramanujan's congruence σ11(n) ≡ τ(n) mod 691. If f is the homomorphism taking (c,d) to c in Z, then the congruence ideal is (691). So the congruence ideal describes the congruences between the forms E12 and Δ.
References
Lenstra, H. W. (1995), "Complete intersections and Gorenstein rings", in
Coates, John (ed.), Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, pp. 99–109,
ISBN1-57146-026-8,
MR1363497,
Zbl0860.13012