125 research outputs found
Algebraicity of higher Green functions at a CM point
In this paper, we investigate the algebraic nature of the value of a higher
Green function on an orthogonal Shimura variety at a single CM point. This is
motivated by a conjecture of Gross and Zagier in the setting of higher Green
functions on the product of two modular curves. In the process, we will study
analogue of harmonic Maass forms in the setting of Hilbert modular forms, and
obtain results concerning the arithmetic of their holomorphic part Fourier
coefficients. As a consequence, we confirm the conjecture of Gross and Zagier
under mild condition on the discriminant of the CM point.Comment: 38 pages, comments welcom
Average CM-values of Higher Green's Function and Factorization
In this paper, we prove an averaged version of an algebraicity conjecture in
\cite{GKZ87} concerning the values of higher Green's function at CM points.
Furthermore, we give the factorization of the ideal generated by such algebraic
value in the spirit of the famous work of Gross and Zagier on singular moduli.Comment: 53 pages, comments welcome
Span of Restriction of Hilbert Theta Functions
In this paper, we study the diagonal restrictions of certain Hilbert theta
series for a totally real field , and prove that they span the corresponding
space of elliptic modular forms when the is quadratic or cubic.
Furthermore, we give evidence of this phenomenon when is quartic, quintic
and sextic.Comment: 19 pages. To appear in PAM
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