Extreme values of central $L$-derivatives and heights of Heegner points

Abstract

Using Soundararajan's resonance method, we obtain extreme values of the central derivatives \[ L^{(r)}(1/2,f\otimesχ_d), \] where $f$ is a normalized newform and ${\chi }_d$ ranges over primitive real characters associated with negative fundamental discriminants. The twisted first moment estimate used in the proof may be of independent interest. Applying Gross--Zagier-type formulas, we obtain correspondingly large Néron--Tate heights of Heegner points on elliptic curves and large absolute values of Beilinson--Bloch heights of Heegner cycles attached to modular forms.