We study Tamagawa numbers and other invariants (especially Tate–Shafarevich sets) attached to commutative and pseudoreductive groups over global function fields. In particular, we prove a simple formula for Tamagawa numbers of commutative groups and pseudoreductive groups. We also show that the Tamagawa numbers and Tate–Shafarevich sets of such groups are invariant under inner twist, as well as proving a result on the cohomology of such groups which extends part of classical Tate duality from commutative groups to all pseudoreductive groups. Finally, we apply this last result to show that for suitable quotient spaces by commutative or pseudoreductive groups, the Brauer–Manin obstruction is the only obstruction to strong (and weak) approximation.

We give a new characterisation of elliptic curves of Shimura type in terms of commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for such curves, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Roth’s theorem is extended to finitely generated field extensions of $\mathbb{Q}$, using Moriwaki’s theory of heights.

We consider the question of simplicity of a $ℂ$-algebra $R$ under the action of its ring of differential operators $D_{R∕ℂ}$. We give examples to show that even when $R$ is Gorenstein and has rational singularities, $R$ need not be a simple $D_{R∕ℂ}$-module; for example, this is the case when $R$ is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano varieties, and our proof proceeds by showing that the tangent bundle of such a variety need not be big. We also give a partial converse showing that when $R$ is the homogeneous coordinate ring of a smooth projective variety $X$, embedded by some multiple of its canonical divisor, then simplicity of $R$ as a $D_{R∕ℂ}$-module implies that $X$ is Fano and thus $R$ has rational singularities.

In this paper, we develop the theory of flag manifolds over a semifield for any Kac–Moody root datum. We show that a flag manifold over a semifield admits a natural action of the monoid over that semifield associated with the Kac–Moody datum and admits a cellular decomposition. This extends the previous work of Lusztig, Postnikov, Rietsch, and others on the totally nonnegative flag manifolds (of finite type) and the work of Lusztig, Speyer, Williams on the tropical flag manifolds (of finite type). As an important consequence, we prove a conjecture of Lusztig on the duality of a totally nonnegative flag manifold of finite type.

We prove a conjecture of Lehmann and Tanimoto about the behaviour of the Fujita invariant (or $a$-constant appearing in Manin’s conjecture) under pull-back to generically finite covers. As a consequence we obtain results about geometric consistency of Manin’s conjecture.

We prove that the ring of Siegel modular forms of weight divisible by $g+n+1$ is isomorphic to the ring of (log) pluricanonical forms on the $n$-fold Kuga family of abelian varieties and certain compactifications of it, for every arithmetic group for a symplectic form of rank . We also give applications to the Kodaira dimension of the Kuga variety. In most cases, the Kuga variety has canonical singularities.