Twisted Borcherds products on Hilbert modular surfaces and their CM values

Details Twisted Borcherds products on Hilbert modular surfaces and their CM values Twisted Borcherds products on Hilbert modular surfaces and their CM values

2005-05-10

arxiv.org/abs/math/0505177v2

Mathematics

Number Theory

30 pages; Theorem 6.6 corrected, references added

Scientific paper

We construct a natural family of rational functions $\tilde\Psi_m$ on a Hilbert modular surface from the classical $j$-invariant and its Hecke translates. These functions are obtained by means of a multiplicative analogue of the Doi-Naganuma lifting and can be viewed as twisted Borcherds products. We then study when the value of $\tilde\Psi_m$ at a CM point associated to a non-biquadratic quartic CM field generates the `CM class field' of the reflex field. For the real quadratic field $\Q(\sqrt{5})$, we factorize the norm of some of these CM values to $\Q(\sqrt 5)$ numerically.

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