Galois sections for abelianized fundamental groups

We consider smooth projective curves C/𝔽 over a finite field and their symmetric squares C (2) . For a global function field K/𝔽, we study the K-rational points of C (2) . We describe the adelic points of C (2) surviving Frobenius descent and how the K-rational points fit there. Our methods also lead to an explicit bound on the number of K-rational points of C (2) satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.

Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (α mod p) is prime to ℓ. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A/F is an elliptic curve with surjective 2-adic representation and α ∈ A(F) with α ∉ 2A(F(A[4])), then the density of p with (α mod p) having odd order is 11/21.

Let $K$ be a number field, $\UX$ be a smooth projective curve over it and $D$ be a reduced divisor on $\UX$. Let $(E,\nabla)$ be a fibre bundle with connection having meromorphic poles on $D$. Let $p_1,...,p_s\in\UX(K)$ and $X:=\UX\setminus\{D,p_1,..., p_s\}$ (the $p_j$'s may be in the support of $D$). Using tools from Nevanlinna theory and formal geometry, we give the definition of $E$--section of type $\alpha$ of the vector bundle $E$ with respect to the points $p_j$; this is the natural generalization of the notion of $E$ function defined in Siegel Shidlowski theory. We prove that the value of a $E$--section of type $\alpha$ in an algebraic point different from the $p_j$'s has maximal transcendence degree. Siegel Shidlowski theorem is a special case of the theorem proved. We give an application to isomonodromic connections.

Let X be a subgroup of a group Y. The \\emphinterval (X, Y) is the set of subgroups of Y that contain X including X and Y. By local class field theory the interval (N K/k K*, k*) contains a finite number of norm groups for any finite extension K of a 𝔭-adic number field k. In the present work we investigate the number of norm groups in the interval (N K/k K*, k*) for a given finite extension K/k of algebraic number fields. We prove that if K/k is an extension of prime degree, or of degree n such that the normal closure of K over k has the Galois group isomorphic to A n or S n , then the interval (N K/k K*, k*) contains only the obvious two norm groups. Also, the interval (N K/k K*, k*) contains a finite number of norm groups for any Galois extension of degree 4, and there are extensions with Galois groups order 8 for which the corresponding intervals contain a finite number of norm groups. The main theorem in our earlier work states that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois extension of even degree that is not a 2-extension, the so-called 2𝔫-\\emphextensions. In the present work we generalize the main theorem to non-Galois 2𝔫-extensions K/k, and determine some subintervals of (N K/k K*, k*) that contain infinitely many norm groups. We then use this theorem to prove that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois 2-extension K/k with the Galois group that either contains an element of order 8 or contains the quaternion group Q8 of order 8, or Q8 is a homomorphic image of the Galois group.

We construct Abel maps for a stable curve X. Namely, for each oneparameter deformation of X to a smooth curve, having regular total space, and each d ≥ 1, we construct by specialization a map αd X : Ẋd → P d X , where Ẋ ⊆ X is the smooth locus, and P d X is the coarse moduli scheme for equivalence classes of degree-d “semibalanced” line bundles on semistable curves having X as a stable model. For d = 1, we show that α1 X extends to a map α1 X : X → P 1 X , and does not depend on the choice of the deformation. Finally, we give a precise description of when α1 X is injective. The theory of Abel maps for smooth curves goes back to the 19th century. In the modern language, let C be a smooth projective curve, and PicC its degree-d Picard variety parametrizing line bundles of degree d on C. For each d > 0 there exists a remarkable morphism, often called the d-th Abel map: Cd −→ PicC (p1, . . . , pd) 7→ OC( ∑ pi). This map has been extensively studied and used in the literature. For d = 1, after the choice of a “base” point on C, it gives the Abel–Jacobi embedding C ↪→ PicC ∼= PicC (unless C ∼= P1). For an interesting historic survey see [K04] or [K05]. What about Abel maps for singular curves? Abel maps were constructed for all integral curves in [AK], and further studied in [EGK00], [EGK02] and [EK05]. In [AK], it is shown that the first Abel map of an integral singular curve is an embedding into its compactified Picard scheme. However almost nothing is known for reducible curves, not even when they are stable. This lack of knowledge appears all the more regrettable because of the importance of stable curves in moduli theory. In the present paper we construct Abel maps for stable curves. As we see it, Abel maps should satisfy the following natural properties. First, they should have a geometric meaning. More explicitly, recall that for a smooth

In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First, we show that, over global fields, quadratic post-critically finite (PCF) polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes’ conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily PCF, and we combine our results with local class field theory to classify quadratic pairs over ${ {\mathbb{Q}}}$ with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other hand, we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence toward Jones’ finite index conjecture.

Let H be the upper half plane and T a discrete subgroup of AutH.Suppose that H mod Y is of finite measure.This work stems from the question whether there is an algebraic interpretation for the moduli of unitary representations of T similar to the case whenWe show that this is indeed the case via the moduli of vector bundles on the compactification of H mod r, provided with some additional structures which we propose to call parabolic structures.The idea of parabolic structures is inspired from A. Weil's work [6,§2, Chapter I, p. 56].Let Xbe a smooth, irreducible, projective curve defined, say, over an algebraically closed field k.By vector bundles on X we understand algebraic vector bundles.DEFINITION 1.Let V be a vector bundle on X and Q G X. Then a quasiparabolic structure of Vat Q is giving a flag on the fibre VQ of Fat Ô, i.e., giving linear subspaces F*VQ of VQ,We call I = (l l9 . . .9 l r ) the type (or flag type) of the quasi-parabolic structure.Let k x = l x -/ 2 , k 2 = l 2 -/ 3 , . . ., fc fWl = /,._! -l r , k r = l r \ then k t are called the multiplicities of the quasi-parabolic structure.DEFINITION 2. Let V be a vector bundle on X and Q G X. Then a parabolic structure of V at Q is giving (i) a quasi-parabolic structure of V at Q; say I = (l t , . . ., l r ) is its type and {k.} its multiplicities, and(ii) constants a = (a 1 , . . ., a n ) called the weights of the parabolic structure such that 0 < a x < a 2 < • • • < ot n < 1 and there are r distinct elements among a, say a' = (o^, . . ., Û£), 0 < a\ < a 2 < • • • < a r < 1, such that OL\ occurs k x times, a 2 occurs k 2 times, . . ., a r occurs k r times among a.We call a. the weight ofF l V Q .Note that l x = « = rkV.Let F, H> be vector bundles on X with quasi-parabolic structures at Q.An isomorphism ƒ: F -• W of vector bundles is said to be a quasi-parabloic isomorphism if the types of V, W at Q are the same and fçiF'Vç) = F 1 WQ (/^:

Equivariant approach to weighted projective curves

The spectrum of Grothendieck monoid: Classifying Serre subcategories and reconstruction theorem

The local lifting problem for actions of finite groups on curves

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit geometric Painlev\'e property as in the regular singular cases proved generally in [8].

A K3 surface over a number field has infinitely many rational points over a finite field extension. For K3 surfaces of degree 2, arising as double covers of P2 branched along a smooth sextic curve, we give a bound for the degree of such an extension. Moreover, using ideas of van Luijk and a surface constructed by Elsenhans and Jahnel, we give an explicit family of K3 surfaces of degree 2 defined over Q with geometric Picard number 1 and infinitely many Q-rational points that is Zariski dense in the moduli space of K3 surfaces of degree 2.

Let K = k(C) be the function field of an algebraic curve C over an algebraically closed ground field k. Let Γ/K be a smooth projective curve of genus g > 0 with a k-rational point O ∈ Γ(K), and let J/K denote the Jacobian variety of Γ/K. Further let (τ, B) be the K/k:-trace of J (see §2 below and).

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> be a smooth projective curve over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℂ</mml:mi></mml:math>. Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>. Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒬</mml:mi></mml:math> be the Quot scheme parameterizing torsion quotients of the vector bundle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>𝒪</mml:mi> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup></mml:math> of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>. In this article we study the nef cone of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒬</mml:mi></mml:math>. We give a complete description of the nef cone in the case of elliptic curves. We compute it in the case when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> very general, in terms of the nef cone of the second symmetric product of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math>. In the case when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> very general, we give upper and lower bounds for the Nef cone. In general, we give a necessary and sufficient criterion for a divisor on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒬</mml:mi></mml:math> to be nef.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.