On Converse Theorems for Hilbert Modular Forms Assuming Unramified Twists

ABSTRACT We prove two results on converse theorems for Hilbert modular forms over totally real fields of degree $r>1$. The first result recovers a Hilbert modular form (of some level) from an L-series satisfying functional equations twisted by all the unramified Hecke characters. The second result assumes both the ‘unramified’ functional equations and an Euler product, and recovers a Hilbert modular form of the expected level predicted by the shape of the functional equations. Our result generalizes the current converse theorems for $\mathrm{GL}_2$ in the case of Hilbert modular forms in that we completely remove the assumptions on ramified twists.