Twisted Borcherds Products on Hilbert Modular Surfaces and Their Cm Values
نویسندگان
چکیده
We construct a natural family of rational functions Ψ̃m 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 Ψ̃m 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( √ 5), we factorize the norm of some of these CM values to Q( √ 5) numerically.
منابع مشابه
Congruence Properties of Borcherds Product Exponents
In his striking 1995 paper, Borcherds [2] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant −d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds’ work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the e...
متن کاملTraces of Singular Values and Borcherds Products
Abstract. Let p be a prime for which the congruence group Γ0(p) ∗ is of genus zero, and j∗ p be the corresponding Hauptmodul. Let f be a nearly holomorphic modular form of weight 1/2 on Γ0(4p) which satisfies some congruence condition on its Fourier coefficients. We interpret f as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite produ...
متن کاملZagier Duality for the Exponents of Borcherds Products for Hilbert Modular Forms
A certain sequence of weight 1/2 modular forms arises in the theory of Borcherds products for modular forms for SL2(Z). Zagier proved a family of identities between the coefficients of these weight 1/2 forms and a similar sequence of weight 3/2 modular forms, which interpolate traces of singular moduli. We obtain the analogous results for modular forms arising from Borcherds products for Hilber...
متن کاملBorcherds products and arithmetic intersection theory on Hilbert modular surfaces
We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Falt...
متن کاملInfinite Products in Number Theory and Geometry
We give an introduction to the theory of Borcherds products and to some number theoretic and geometric applications. In particular, we discuss how the theory can be used to study the geometry of Hilbert modular surfaces.
متن کامل