On the congruence of modular forms.(English)Zbl 0615.10035
Number theory, Semin. New York 1984/85, Lect. Notes Math. 1240, 275-280 (1987).
[For the entire collection seeZbl 0605.00005.]
The author studies congruences between modular forms with the aim of deciding when two modular forms with algebraic integer coefficients are congruent modulo a prime \(\lambda\). To be more precise, let R be the ring of integers in a number field F, and let \(\lambda\) be a prime ideal of R. Let \(f=\sum c_ nq^ n\) be the q-expansion of a weight k modular form f on some congruence subgroup \(\Gamma\) of \(SL_ 2\). Define \(ord_{\lambda}(f)\) to be inf\(\{\) \(n: \lambda\nmid c_ n\}\) with \(ord_{\lambda}(f)=\infty\) if \(c_ n\equiv 0\) (mod \(\lambda)\) for all n. The author shows that if \(f,g\in M_ k(\Gamma,R)\), and \(ord_{\lambda}(f-g)>k[\Gamma (1) : \Gamma]/12\) then \(f\equiv g\) (mod \(\lambda)\). The theorem is first proved for \(\Gamma =\Gamma (1)\). The case of an arbitrary subgroup \(\Gamma\) is treated by modifying f-g in such a way that the problem is reduced to knowing the result, over a certain extension of F, for the group \(\Gamma\) (1).
The author also gives a somewhat sharper result for newforms in \(S_ k(\Gamma_ 0(N),\chi,R)\) when N is square-free. This follows from results ofT. Asai [J. Math. Soc. Japan 28, 48-61 (1976;Zbl 0313.10026)] together with the theorem described above.
The author studies congruences between modular forms with the aim of deciding when two modular forms with algebraic integer coefficients are congruent modulo a prime \(\lambda\). To be more precise, let R be the ring of integers in a number field F, and let \(\lambda\) be a prime ideal of R. Let \(f=\sum c_ nq^ n\) be the q-expansion of a weight k modular form f on some congruence subgroup \(\Gamma\) of \(SL_ 2\). Define \(ord_{\lambda}(f)\) to be inf\(\{\) \(n: \lambda\nmid c_ n\}\) with \(ord_{\lambda}(f)=\infty\) if \(c_ n\equiv 0\) (mod \(\lambda)\) for all n. The author shows that if \(f,g\in M_ k(\Gamma,R)\), and \(ord_{\lambda}(f-g)>k[\Gamma (1) : \Gamma]/12\) then \(f\equiv g\) (mod \(\lambda)\). The theorem is first proved for \(\Gamma =\Gamma (1)\). The case of an arbitrary subgroup \(\Gamma\) is treated by modifying f-g in such a way that the problem is reduced to knowing the result, over a certain extension of F, for the group \(\Gamma\) (1).
The author also gives a somewhat sharper result for newforms in \(S_ k(\Gamma_ 0(N),\chi,R)\) when N is square-free. This follows from results ofT. Asai [J. Math. Soc. Japan 28, 48-61 (1976;Zbl 0313.10026)] together with the theorem described above.
Reviewer: S.Kamienny
MSC:
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
| 11F11 | Holomorphic modular forms of integral weight |
