Products of Hecke eigenforms (Q2581383)

This paper considers the question: ``When is the product of two Hecke eigenforms on \(\Gamma_0(p)\) also an eigenform?'' The answer to this type of question is believed to be ``very rarely.'' For instance, in level one we have identities of the form \(E_4\cdot E_4=E_8\) (where \(E_k\) denotes the weight \(k\) Eisenstein series with value \(1\) at the cusp), which are due to the small dimensions of the vector spaces in question. In [Number Theory in Progress, Vol. 2, 737--741 (1997; Zbl 0953.11017)], \textit{W. Duke} showed that the \(16\) known such identities are the only ones occurring on \(\mathrm{SL}_2(\mathbb{Z})\). These identities can be ``lifted'' to \(\Gamma_0(p)\), giving rise to so-called ``oldform solutions''. The author of the paper under review shows that when \(p\geq 7\) and \(f,g\) are non-cuspidal eigenforms of weights \(k\), resp. \(\ell\), such that \(fg\) is also an eigenform, then we are in one of two cases: 1. \((k,\ell)\in\{(4,6),(4,10),(6,4),(6,8),(8,6),(10,4)\}\) and \[ (f(z),g(z))\in\{(E_k(z),E_\ell(z)),(E_k(pz),E_\ell(pz))\}; \] 2. \((k,\ell)=(4,4)\) and there is a one-parameter family of solutions \[ (E_4(z)+bE_4(pz))(E_4(z)-bE_4(pz))=E_8(z)-b^2E_8(pz). \] The second result is that if \(p\geq \), \(f\) an eigenform of weight \(k\) and \(g\) a cusp form of weight \(\ell\) such that \(h(z)=f(z)g(z)\) is also an eigenform, then either \(h(z)\) is an oldform solution or \[ (p,k,\ell)\in\{(11,4,2),(7,2,6),(7,4,4),(5,2,4),(5,2,8),(5,4,4),(5,4,6),(5,6,4)\}, \] in which case there is a unique solution. Both results are proved by elementary methods and are based on the fact that any non-cuspidal weight \(k\) eigenform on \(\Gamma_0(p)\) is a linear combination of \(E_k(z)\) and \(E_k(pz)\) (a good reference for this is Theorem 4.5.2 of [\textit{F. Diamond} and \textit{J. Shurman}, A first course in modular forms, GTM 228, Berlin: Springer (2005; Zbl 1062.11022)]).