Fourier coefficients of noncongruence cuspforms (Q2893276)

Let \(\Gamma\) be a finite index subgroup of \(\mathrm{SL}_2(\mathbb{Z})\), and \(f(\tau)\) a weight \(k\) weakly holomorphic modular form for \(\Gamma\) with algebraic Fourier coefficients. The unbounded denominators conjecture states that the Fourier coefficients of \(f(\tau)=\sum a(n)q^{n/\mu}\) have bounded denominators if and only if \(f(\tau)\) is a form for a congruence subgroup.NEWLINENEWLINEIn the paper under the review, the authors prove that the conjecture is true for cusp forms in \(S_k(\Gamma), k>1\), when a) the modular curve for \(\Gamma\) has a model over \(\mathbb{Q}\) such that the cusp at \(\infty\) is \(\mathbb{Q}\)-rational; b) \(S_k(\Gamma)\) is one dimensional.NEWLINENEWLINEUnder these assumptions, the result of \textit{A. J. Scholl} [Invent. Math. 79, 49--77 (1985; Zbl 0553.10023)] implies that there is a normalized newform \(g(\tau)=\sum_{n=1}^\infty b(n)q^n\) of some level \(N\), weight \(k\) and character \(\chi\), and an integer \(M\), such that for every prime \(p>M\) the following (Atkin and Swinnerton-Dyer) congruence relation holds NEWLINE\[NEWLINEa(np)-b(p)a(n)+\chi(p)p^{k-1}a(n/p)\equiv 0 \pmod{p^{(k-1)(1+ord_pn)}} .NEWLINE\]NEWLINE Using the Selberg upper bound for the Fourier coefficients of cusp forms, the congruence relation above, and the assumption that \(a(n)'s\) have bounded denominators, the authors prove that for given \(m \in \mathbb{N}\), there is an integer \(P(m)\) such that if \(p_1, p_2, \dots, p_r\) are distinct primes greater than \(P(m)\) and coprime to \(m\) then NEWLINE\[NEWLINEa(mp_1^{e_1}\cdots p_r^{e_r})=a(m)b(p_1^{e_1}\cdots p_r^{e_r}).NEWLINE\]NEWLINE The result above implies that a certain twist of \(f(\tau)\) is a modular form for a congruence subgroup which implies that \(f(\tau)\) is also a modular form for a congruence subgroup.