We prove the Gross–Zagier–Zhang formula over global function fields of arbitrary characteristics. It is an explicit which relates Néron-Tate heights CM points on abelian varieties and central derivatives associated quadratic base change L-functions. Our proof based arithmetic variant a relative trace identity Jacquet. This approach proposed by Zhang. apply our results to Birch Swinnerton–Dyer c...

We consider the Zhang sandpile model in one-dimension (1D) with locally conservative (or dissipative) dynamics and examine its total energy fluctuations at external drive time scale. The bulk-driven system leads to Lorentzian spectra, a cutoff $T$ growing linearly size $L$. show $1/f^{\alpha}$ behavior $\alpha \sim 1$ for boundary drive, varies non-linearly. For local dynamics, shows power-law ...