Can the product of two Hecke eigenforms be another Hecke eigenform? This is a question independently considered by Duke and Ghate for elliptic modular forms of full level, in which case they proved that there are exactly 16 of such eigenform product identities. Later, extending to general level and allowing Dirichlet characters, Johnson proved that there are exactly 61 such identities. In this talk, we show that over all totally real number fields of a fixed degree and all Hilbert Hecke eigenforms of full level, there are at most finitely many such identities. This is joint with Joshi and with You.
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