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Deformations of Theta Integrals and A Conjecture of Gross-Zagier
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In this paper, we complete the proof of the conjecture of Gross and Zagier concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, which is constructed as the Doi-Naganuma theta lift of a deformed theta integral on hyperbolic space.
Forward citations
Cited by 2 Pith papers
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Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields
Twisted theta integrals over non-Galois quartic CM fields equal Doi–Naganuma lifts of Hecke's integral, making twisted CM values of Borcherds forms algebraic multiples of logarithms of units.
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Hilbert Eisenstein series as Doi-Naganuma lift
Incoherent Hilbert Eisenstein series over real quadratic fields are realized as Doi-Naganuma theta lifts, yielding explicit Rankin-Selberg L-functions and a non-unit theorem for Borcherds products at CM points.
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