Congruences#
- CongruenceGroup(M1, M2, prec): ModFrm, ModFrm, RngIntElt -> GrpAb#
A group \(C\) that measures all possible congruences (to precision \(prec\)) between some modular form in \(M_1\) and some modular form in \(M_2\). The group \(C\) is defined as follows. Let \(W_1\) be the finite-rank \({\mathbb{Z}}\)-module \(q\)-exp\((M_1)\cap {\mathbb{Z}}[[q]]\) and let \(W_2\) be \(q\)-exp\((M_2)\cap {\mathbb{Z}}[[q]]\). Let \(V\) be the saturation of \(W_1 + W_2\) in \({\mathbb{Z}}[[q]]\). Then \(C=V/(W_1+W_2)\).
- CongruenceGroupAnemic(M1, M2, prec): ModFrm, ModFrm, RngIntElt -> GrpAb#
Analogous to
CongruenceGroup, but now considering congruences that hold for all q-expansion coefficients \(a_n\) with \(n\) coprime to the levels of both \(M1\) and \(M2\) (rather than for all q-expansion coefficients).
- Example: Congruences (ex-71c3ed)#
We verify that the newform corresponding to the first elliptic curve of rank \(2\) is congruent modulo \(5\) to some Galois-conjugate newform corresponding to the winding quotient of \(J_0(389)\) (there is also a congruence modulo \(2\) to some conjugate form).
> M := ModularForms(Gamma0(389),2); > f := Newform(M,1); > Degree(f); 1 > g := Newform(M,5); > Degree(g); 20 > CongruenceGroup(Parent(f),Parent(g),30); Abelian Group isomorphic to Z/20 Defined on 1 generator Relations: 20*$.1 = 0
The congruence can be seen directly by computing the reductions of \(f\) and \(g\) modulo \(5\):
> fmod5 := Reductions(f,5); > gmod5 := Reductions(g,5); // takes a few seconds. > #gmod5; 7 > #fmod5; 1 > [gbar : gbar in gmod5 | #gbar eq 1]; [ [* q + 4*q^2 + q^3 + 4*q^4 + q^5 + 4*q^6 + O(q^8) *], [* q + 3*q^2 + 3*q^3 + 2*q^4 + 2*q^5 + 4*q^6 + O(q^8) *] ] > fmod5[1][1]; q + 3*q^2 + 3*q^3 + 2*q^4 + 2*q^5 + 4*q^6 + O(q^8)