On the Tits indices of absolutely almost simple algebraic groups over local and global fields

On analogs of Cassels–Tate's exact sequence for connected reductive groups and Brauer-Manin obstruction for homogeneous spaces over global function fields

On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I

Let K be a field, Br(K) its Brauer group. If L/K is a field extension, then the relative Brauer group Br(L/K) is the kernel of the restriction map resL/K : Br(K)→ Br(L). Relative Brauer groups have been studied by Fein and Schacher. Every subgroup of Br(K) is a relative Brauer group Br(L/K) for some extension L/K, and the question arises as to which subgroups of Br(K) are algebraic relative Brauer groups, i.e. of the form Br(L/K) with L/K an algebraic extension. For example if L/K is a finite extension of number fields, then Br(L/K) is infinite, so no finite subgroup of Br(K) is an algebraic relative Brauer group. In [1] the question was raised as to whether or not the n-torsion subgroup Brn(K) of the Brauer group Br(K) of a field K is an algebraic relative Brauer group. For example, if K is a (p-adic) local field, then Br(K) ∼= Q/Z, so Brn(K) is an algebraic relative Brauer group for all n. A counterexample was given in [1] for n = 2 and K a formal power series field over a local field. For global fields K, the problem is a purely arithmetic one, because of the fundamental local-global description of the Brauer group of a global field. In particular, for a Galois extension L/K of global fields, if the local degree of L/K at every finite prime is equal to n, and is equal to 2 at the real primes for n even, then Br(L/K) = Brn(K). In [1], it was proved that Brn(Q) is an algebraic relative Brauer group for all squarefree n. In [2], the arithmetic criterion above was verified for any number field K Galois over Q and any n prime to the class number of K, so in particular, Brn(Q) is an algebraic relative Brauer group for all n. In [3], Popescu proved that for a global function field K of characteristic p, the arithmetic criterion holds for n prime to the order of the non-p part of the Picard group of K. In this paper we settle the question completely, by verifying the arithmetic criterion for all n and all global fields K. In particular, the n-torsion subgroup of the Brauer group of K is an algebraic relative Brauer group for all n and all global fields K. The proof, an extension of the ideas in [2], reduces to the case n a prime power `. We first carry out the proof for number fields K. The proof for the function field case when ` 6= char(K) is essentially the same as the proof in the number field case. The proof for ` = char(K) appears in [3].

So far we have been working with the polynomial ring A inside the rational function field k = F(T). In this section we extend our considerations to more general function fields of transcendence degree one over a general constant field. This process is somewhat like passing from elementary number theory to algebraic number theory. The Riemann-Roch theorem is the fundamental result needed to accomplish this generalization. We will give a proof of this fundamental result in Chapter 6. In this chapter we give the basic definitions, state the theorem, and derive a number of important corollaries. After this is accomplished, attention will be shifted to function fields over a finite constant field. Such fields are called global function fields. The other class of global fields are algebraic number fields. All global fields share a great number of common features. We introduce the zeta function of a global function field and explore its properties. The Riemann hypothesis for such zeta functions will be explained in some detail, and we will derive several very important consequences, among others an analogue for the prime number theorem for arbitrary global function fields. A proof of the Riemann hypothesis will be given in the appendix. In this chapter we will prove a weak version. This is enough to yield the analogue of the prime number theorem, albeit with a poor error term. In later chapters we will also explore L-functions associated to global function fields - both Hecke L-functions (generalizations of Dirichlet L-functions) and Artin L-functions.KeywordsZeta FunctionFunction FieldGlobal FunctionAlgebraic FunctionRiemann HypothesisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper shows that the group is boundedly elementary generated for and the ring of algebraic integers in a global function field. Contrary to previous statements for number fields and earlier statements for global function fields, the bounds proven in this paper for elementary bounded generation are independent of the underlying global function field and only depend on the integer . Combining our main result with earlier results, we further establish that elementary bounded generation always has bounds independent from the global field in question, only depending on .

In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 11 elements. (2) For del Pezzo surfaces of degree 4 defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 13 elements. (3) Weak approximation holds for cubic hypersurfaces of dimension at least 10 defined over a global function field of characteristic not equal to 2, 3, 5 or a purely imaginary number field.

Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets H1(K,G) for all reductive K-groups G in a functorial way.

Introduction. Around 1980, Galovich and Rosen (cf. [GR1] and [GR2]) computed the index of cyclotomic units in the full group of units in a cyclotomic function field over a rational function field over a finite field. Later, Hayes [H1] and Oukhaba [O] obtained a few index formulae of the elliptic units in some special extensions of the global function fields with some restrictions on the infinite prime. Feng and Yin [FY] constructed the maximal independent systems of cyclotomic units in finite abelian extensions over a rational function field. Recently Shu [S] extended the result in [GR1] to the global function fields with degree one infinite prime. In the Mini-Conference on the Arithmetic of Function Fields held at Brown University in April 1996, Yin announced a result along this line which extended his work in [Y] by removing the restriction on the degree of the infinite prime. This result seems to be the best one concerning the cyclotomic unit index in the sense that the base field can be any global function field. In this paper we construct maximal independent systems of units in an abelian extension over such a global function field. Notations and terminology are standard if not explained. Specifically, • k: a global function field with a constant field Fq of q elements. • ∞: a fixed infinite prime. • A: the ring of the functions in k which are holomorphic away from ∞. • M∞: the set of integral ideals of A. • e: the unit ideal of A. • P: the set of k-primes (k-places). • kv: the local field over k completed at any v ∈ P. For any m ∈M∞, • Pm := {p ∈ P : (p,m) = 1}. • Mm := {b ∈M∞ : (b,m) = 1}. • mv (resp. Av): the completion of m (resp. A) in kv.

We give some new formulas via some exact sequences for computing an obstruction to the weak approximation on non-abelian cohomology sets and homogeneous spaces over global fields, with stabilizers belonging to some class of non-connected subgroups. As a consequence, we show that the Brauer–Manin obstruction to the weak approximation for such spaces is the only one. Along the way, we show that the Brauer–Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global function fields with stabilizers belonging to a certain class of non-necessarily connected groups are the only ones, extending some of Borovoi’s results obtained for number fields in this regard.

We show that the Brauer–Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global function fields with connected reductive stabilizers are the only ones, extending some of Borovoi’s results (and thus also proving a partial case of a conjecture of Colliot-Thelene) in this regard. Along the way, we extend some perfect pairings and an important local-global exact sequence (an analog of a Cassels–Tate’s exact sequence) proved by Sansuc for connected linear algebraic groups defined over number fields, to the case of connected reductive groups over global function fields and beyond.

We determine the distribution of discriminants of wildly ramified elementary-abelian extensions of local and global function fields in characteristic p p . For local and rational function fields, we also give precise formulae for the number of elementary-abelian extensions with a fixed discriminant divisor, which describe a local-global principle.

Diophantine equations over global function fields IV: S-unit equations in several variables with an application to norm form equations

The problem of writing a totally positive element as a sum of squares has a long history in mathematics, going back to Bachet and Lagrange. While for some specific rings (like integers or polynomials over the rationals), there are known methods for decomposing an element into a sum of squares, in general, for many other important rings and fields, the problem is still widely open. In this paper, we present an explicit algorithm for decomposing an element of an arbitrary global field (either a number field or a global function field) into a sum of squares of minimal length.

We formulate a general problem: Given projective schemes and over a global field K and a K‐morphism η from to of finite degree, how many points in of height at most B have a pre‐image under η in ? This problem is inspired by a well‐known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when and is a prime degree cyclic cover of . Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.