We state and prove a formula for a certain value of the Goss L-function of a Drinfeld module. This gives characteristic-p-valued function field analogues of the class number formula and of the Birch and Swinnerton-Dyer conjecture. The formula and its proof are presented in an entirely selfcontained fashion.

We show that the module of integral points on a Drinfeld module satisfies an analogue of Dirichlet's unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated sub-module of the module of integral points. We use the results to give a precise formulation of a conjectural analogue of the class number formula.

Abstract. Recently the second author has associated a finite Fq[T ]-module

We show that the Looijenga-Lunts-Verbitsky Lie algebra acting on the cohomology of a hyperkähler variety is a derived invariant. We use this to give upper bounds for the representation of the group of derived autoequivalences on the cohomology of a hyperkähler variety. For certain Hilbert squares of K3 surfaces, we show that the obtained upper bound is close to being sharp.

We prove a function field analogue of the Herbrand-Ribet theorem on cyclotomic number fields. The Herbrand-Ribet theorem can be interpreted as a result about cohomology with μ p -coefficients over the splitting field of μ p , and in our analogue both occurrences of μ p are replaced with the p-torsion scheme of the Carlitz module for a prime p in F q [t].