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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