Related papers: On the birational section conjecture over finitely…

This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the…

We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…

The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from $2$ this theorem is extended here to function…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

In this paper, we elaborate the theory of exceptional hereditary curves over arbitrary fields. In particular, we study the category of equivariant coherent sheaves on a regular projective curve whose quotient curve has genus zero and prove…

We prove that the group of birational automorphisms of a geometrically irreducible algebraic surface over a finite field is Jordan. We show that the analogous statement fails in higher dimensions. Finally, we prove that groups of birational…

We prove that the Tate conjecture in codimension $1$ over a finitely generated field follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due to de Jong--Morrow over $\mathbf{F}_p$ and…

Given a finite field k of characteristic at least 5, we show that the Tate conjecture holds for K3 surfaces defined over the algebraic closure of k if and only if there are only finitely many K3 surfaces over each finite extension of k.

We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for Q-factorial surfaces and for log canonical surfaces. Moreover, in the…

In this paper the number of $\mathbb{F}_q$-isomorphism classes of Legendre elliptic curves over the finite fields $\mathbb{F}_q$ is enumerated.

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…

We show that Generic Green's conjecture holds for generic binary curves, through a detailed analysis of the family of scrolls containing fixed rational normal curves.

On a reasonable class of domains in $\CC^n$, we characterize those holomorphic functions which continue analytically past the boundary. Then we give some applications of this result to holomorphic mappings. In addition, some new results…

We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We…

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed…

A birationally liftable Galois section s of a hyperbolic curve X/k over a number field k yields an adelic point x(s) in the smooth completion of X. We show that x(s) is X-integral outside a set of places of Dirichlet density 0, or s is…

We prove special cases of a general conjecture: If an invertible field theory admits a projectively topological boundary theory, then it has finite order in the abelian group of invertible field theories. One can substitute `gapped' for…

A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant…

We study the problem of tiling and packing in vector spaces over finite fields, its connections with zeroes of classical exponential sums, and with the Jacobian conjecture