Hilbert modular forms are one generalization of the concept of modular form to functions of more than one complex variable. (Another such generalization are Siegel modular forms.)
Just as modular forms are, locally, functions on the moduli space of elliptic curves over the complex numbers, so Hilbert modular forms are locally functions on moduli spaces of something like complex abelian varieties with real multiplication and polarization module (Goren, p. 49).
Hilbert modular forms have a slick equivalent formulation as adelic automorphic forms. (…)
Jan Hendrik Bruinier, Hilbert modular forms and their applications (arXiv:0609763)
Eyal Z. Goren, Lectures on Hilbert Modular Varieties and Modular Forms
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