The structure of the Tate–Shafarevich groups of a class of elliptic curves over global function fields is determined. These are known to be finite abelian groups and hence they are direct sums of finite cyclic groups where the orders of these cyclic components are invariants of the Tate–Shafarevich group. This decomposition of the Tate–Shafarevich groups into direct sums of finite cyclic groups depends on the behaviour of Drinfeld–Heegner points on these elliptic curves. These are points analogous to Heegner points on elliptic curves over the rational numbers.

Linear diffusions which are not far from Bessel diffusions are considered. Specifically, the regular variation of Feller’s canonical measure is interpreted in terms of the drift coefficient. As an application, the asymptotic behavior of the transition probability at large times is discussed.

In this article, we propose a geometric analogue of Dirichlet’s unit theorem on arithmetic varieties; that is, if $X$ is a normal projective variety over a finite field and $D$ is a pseudo-effective $\mathbb{Q}$-Cartier divisor on $X$, does it follow that $D$ is $\mathbb{Q}$-effective? We also give affirmative answers on an abelian variety and a projective bundle over a curve.

Extending previous methods, we prove three structure theorems for vector-valued modular forms, where two correspond to $4$-dimensional cases (two Hermitian modular groups, one belonging to the field of Eisenstein numbers and the other to the field of Gaussian numbers) and one to a $6$-dimensional case (a quaternionic modular group).

We classify complete biharmonic surfaces with parallel mean curvature vector field and nonnegative Gaussian curvature in complex space forms.

We investigate quantization coefficients for probability measures $\mu$ on limit sets, which are generated by systems $\mathcal{S}$ of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization coefficients for infinite systems has significant differences from the finite case. One of these differences is the lack of finite maximal antichains, and another is the fact that the set of contraction ratios has zero infimum; another difference resides in the specific geometry of $\mathcal{S}$ and of its noncompact limit set $J$. We prove that, for each $r\in (0,\infty )$, there exists a unique positive number ${\kappa }_r$, so that for any $\kappa <{\kappa }_r<\kappa \text{'}$, the $\kappa$-dimensional lower quantization coefficient of order $r$ for $\mu$ is positive, and we give estimates for the $\kappa \text{'}$-upper quantization coefficient of order $r$ for $\mu$. In particular, it follows that the quantization dimension of order $r$ of $\mu$ exists, and it is equal to ${\kappa }_r$. The above results allow one to estimate the asymptotic errors of approximating the measure $\mu$ in the $L_r$-Kantorovich–Wasserstein metric, with discrete measures supported on finitely many points.