Mathematical Research Letters

Volume 19 (2012)

Number 2

The derivative of an incoherent Eisenstein series II

Pages: 273 – 289

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n2.a2

Author

Hui Xue (Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, U.S.A.)

Abstract

Let $K$ be an imaginary quadratic field with the norm map$\N$. Let $(K,-\kappa\N)$ be a binary quadratic form with$\kappa$ a positive rational number. In this paper, undersome minor conditions on $\kappa$, we first construct anincoherent Eisenstein series (in the sense of Kudla)associated to $(K,-\kappa\N)$. Then we study its derivativeat the center of symmetry $s=0$, and show that eachnon-constant Fourier coefficient of the derivative can beinterpreted as the degree of certain zero-dimensionalschemes. This result together with our previous work givesa complete answer to a question raised by Kudla–Rapoport–Yang.

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