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Congruence relations between modular forms. (English) Zbl 0575.10024
Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 503-514 (1984).
The paper under review continues the work of H. Hida [e.g. Invent. Math. 63, 225–261 (1981; Zbl 0459.10018)], the author [Invent. Math. 71, 193–205 (1983; Zbl 0508.10018)] and others concerning the notion of congruence primes in the theory of elliptic modular forms. One starts with a decomposition \(S_ k=X\oplus Y\) of the space of cusp forms of weight \(k\) with respect to a congruence subgroup \(\Gamma\). This decomposition should be stable under the action of Hecke operators and both \(X\) and \(Y\) should be generated by their intersections with \(S_ k({\mathbb Z})\). A prime \(p\) is called a congruence prime, if there exist \(f\in X\cap S_ k({\mathbb Z})\) and \(g\in Y\cap S_ k({\mathbb Z})\) with \(f\equiv g\bmod p\) and \(f\not\equiv 0 \bmod p\).
The author studies the particular example \(k=2\), \(\Gamma =\Gamma_ 0(NM)\), \((N,M)=1\), \(X=\) subspace of old forms associated with \(\Gamma_ 0(N)\) and \(Y=X^{\perp}\). It is shown that the explicit calculation of congruence primes in these examples leads into problems concerning certain finite subgroups of Jacobians of modular curves, among them the so-called Shimura subgroup of \(J_ 0(N)\).
[For the entire collection see Zbl 0553.00001.]

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight