The \(\ell\)-adic representations attached to a certain noncongruence subgroup. (English) Zbl 0647.10022

We consider the (one-dimensional) space of holomorphic cusp forms of weight 4 on a certain subgroup \(\Gamma_{711}\) of \(\mathrm{PSL}_2(\mathbb{Z})\), which is not a congruence subgroup. Associated to this space is a compatible system of 2-dimensional \(\ell\)-adic representations of \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\). The relation between the Galois representations and the Fourier coefficients is expressed by the congruences of A. O. L. Atkin and H. P. F. Swinnerton-Dyer [Proc. Symp. Pure Math. 19, 1–25 (1971; Zbl 0235.10015)], which were established by the author [Invent. Math. 79, 49–77 (1985; Zbl 0553.10023)]. In this paper we show that the \(L\)-series associated to this system of representations is in fact the \(L\)-series of a certain cusp form of weight 4 on the congruence subgroup \(\Gamma_0(14).\)
The method is to compare the 2-adic representations associated to the two different cusp forms, and show that they are isomorphic. To do this we calculate the trace of Frobenius elements for a small number of primes in the two representations. For the congruence modular form these occur in the Fourier expansion, and for the non-congruence form they can be determined using the Lefschetz formula in \(\ell\)-adic cohomology. A result of Serre (based on a method of Faltings; see R. LivnĂ©, Contemp. Math. 67, 247–261 (1987; Zbl 0621.14019)] is then be used to show that the representations are isomorphic.
Reviewer: A. J. Scholl


11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight
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