On the Section Conjecture over Function Fields and Finitely Generated Fields

Given a smooth geometrically connected curve C over a field k and a smooth commutative group scheme G of finite type over the function field K of C, we study the Tate–Shafarevich groups given by elements of \(H^1(K,G)\) locally trivial at completions of K associated with closed points of C. When G comes from a k-group scheme and k is a number field (or k is a finitely generated field and C has a k-point), we prove finiteness of generalizing a result of Saïdi and Tamagawa for abelian varieties. We also give examples of nontrivial in the case when G is a torus and prove other related statements.

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicated to describe the position of decomposition groups of points at the boundary of the scheme \({{\rm Spec}\, \mathcal{O}_{K, S}}\), where K is a number field and S a set of primes of K, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally to the fundamental group \({\pi_1({\rm Spec}\, \mathcal{O}_{K, S})}\) : the location of decomposition groups of boundary points inside it, the p-part of the cyclotomic character, the number of points on the boundary of all finite étale covers, etc. Under a certain finiteness hypothesis on Tate–Shafarevich groups with divisible coefficients, one can reconstruct all these quantities simply from the fundamental group.

We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. The main theorems of this paper (5.5 and 5.6) establish that, for a degree 2 map, semi-stability in the Geometric Invariant Theory sense on the space of self maps, implies minimality of the resultant. We prove the singular reduction of a semi-stable presentation coincides with the simple bad reduction (Theorem 4.1). Given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes map has unstable bad reduction (Proposition 4.6). Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample (Example 3.1) to a simple dynamical analog to Theorem 1.1.

Classical covering systems, introduced by Erdo’s, consist of congruences ai (mod ni) whose union covers all integers. Despite extensive work on their structural and extremal properties, little is known about analogues in algebraic settings. In this paper, we develop a unified framework for covering systems over algebraic domains, focusing on the ring of integers of a number field and the polynomial ring [x]. We define algebraic covering systems in both environments and establish necessary norm and degree-based conditions for full coverage and demonstrate that restricted families of ideals or polynomial moduli cannot yield coverings unless their reciprocal norm sums exceed explicit thresholds. We further provide structural examples, and counterexamples illustrating how factorization patterns, prime splitting, and residue structure influence covering behavior. Our results show that polynomial rings admit sharper and more uniform obstruction criteria than number fields, while number-field coverings exhibit arithmetic constraints governed by prime ideal decomposition.

We present a randomized polynomial-time algorithm to generate an ideal and its factorization uniformly at random in a given number field. We do this by generating a random integer and its factorization according to the distribution of norms of ideals at most N N in the given number field. Using this randomly generated norm, we can produce a random factored ideal in the ring of algebraic integers uniformly at random among ideals with norm up to N N , in randomized polynomial time. We also present a variant of this algorithm for generating random factored ideals in function fields.

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

ABC implies a Zsigmondy principle for ramification

On Roth's theorem. The celebrated theorem of Roth, together with its generalizations given by Mahler and Ridout, gives a lower bound for the degree of approximation of one or more algebraic numbers with respect to a fixed set of valuations by elements of a fixed number field. An analogous result holds for function fields in characteristic zero. In this paper we do the following: (1) generalize Roth's theorem to the case of fields with a product formula in characteristic zero, removing any technical hypothesis from a previous result of Lang: (2) give a unified proof of Roth's theorem in the number field and function field cases; (3) provide a quantitative version of the general Roth's theorem, extending, even in the number field case, previous results of Bombieri and Van Der Poorten.

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese 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.

In 1973, Drinfeld introduced the notion of elliptic modules, which is now known as Drinfeld modules. After that the analogies between number fields and function fields have many interesting new aspects. Drinfeld modular function theory is one of these. Drinfeld modular functions were studied by several mathematicians and known to have many properties analogous to those of classical elliptic functions, such as the generators of modular function fields, Galois groups between them. ([5],[8],[9]). In the first part of this note, we establish some more properties of Drinfeld modular functions in analogy with those obtained by Shimura. In [12], Shimura proved his exact-sequence and his reciprocity law. Lang proved the exact-sequence another way in [11] using the isogeny theory. Shimura's proof of the reciprocity law is not easy. For example he used the parametrizations of the models of a modular function field over Q. In [11], Lang avoided Shimura's method using the decomposition gorups which is well-known in algebraic number theory. In this article, we will follow Shimura's method to prove the exact-sequence, and Lang's method to prove the reciprocity law in the function field case. In the second part, we go on to study two variable Drinfeld modular functions in analogy with two variable elliptic functions studied by Berndt. ([!]). In [2], he also gerneralized Shimura's exact-sequence and his reciprocity law corresponding to this extended modular function fields. We discuss the analogies of these in the Drinfeld setting. 1. Definitions and basic facts Let Λ=/yT], k=Fq(T k^ be the completion of k at 00= (£), and C the completion of the algebraic closure of k^. Then C has an absolute value extending that of k^. By an ^-lattice in C, we mean a projective Λ-submodule A of C which is discrete in the topology of C. A meromorphic function /on C is said to be even if/(μz)=/(z) for every μeF*. A meromorphic function /on C is

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

We classify all one-class genera of admissible lattice chains of length at least 2 in hermitian spaces over number fields. If L is a lattice in the chain and \(\mathfrak {p} \) the prime ideal dividing the index of the lattices in the chain, then the \(\{ \mathfrak {p} \} \)-arithmetic group \(\mathrm {Aut} (L_{ \{ \mathfrak {p} \} }) \) acts chamber transitively on the corresponding Bruhat-Tits building. So our classification provides a step forward to a complete classification of these chamber transitive groups which has been announced 1987 (without a detailed proof) by Kantor, Liebler and Tits. In fact we find all their groups over number fields and one additional building with a discrete chamber transitive group.

For cuspidal automorphic forms on function fields and for cohomological cuspidal automorphic forms on number fields we give estimates for the p-adic valuations of eigenvalues of Hecke operators. In the case of number fields, these estimates correspond to Katz-Mazur estimates by the Langlands conjectures.

In the Langlands program over number fields, automorphic representations and Galois representations are placed into correspondence, using the cohomology of Shimura varieties as an intermediary. Over a function field, the appropriate intermediary is a moduli space of shtukas. We review the shtukas and introduce their local analogues, which play a similar role in the Langlands program over a p p -adic field. Along the way we construct the Fargues-Fontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the contribution of Fargues-Scholze.