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

Using Soundararajan’s resonance method, we obtain extreme values of the central derivatives $$L^{(r)}(1\mathrm{/}2,f\otimes {\chi }_d),$$ 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.