Abstract. We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over Q is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.
We prove that there are only finitely many isomorphism classes of smooth, hyperbolic curves over an algebraic closure of the finite prime field Fp, whose (tame) fundamental group is isomorphic to a prescribed profinite group. This is a generalization of partial results by Pop, Saïdi and Raynaud. The key ingredient of the proof is Raynaud's theory of theta divisors. In course of the proof, we also obtain some results concerning gonalities of coverings of curves and concerning the infinitesimal Torelli problem for generalized Prym varieties, which are applicable to arbitrary (not necessarily positive) characteristic and may be of some interest independent of the study of fundamental groups of curves in positive characteristic.Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:09:02 EDT 2015 for download from IP 130.74.92.202. X of degree not divisible by char(k), such that X is not almost elliptic. Now, we return to the outline of proof of (0.3) (in the case g ≥ 2, n = 0). It is not difficult to reduce the problem to the case where S is the spectrum of k 0 [[t]] and s (resp. t) is the closed (resp. generic) point of S. First, we denote by J 1,snc,t the maximal abelian subvariety of the Jacobian variety J 1,t of X 1,t which does not admit a non-trivial constant quotient abelian variety. Note that J 1,snc,t may be trivial. (For example, this happens if J 1,t is supersingular.) Let J 1,snc,s denote the image in J 1,s of the special fiber of the Néron model of J 1,snc,t . Then, assuming that (0.4) holds, we can prove (see (7.1)) that, for each irreducible component Θ i,s of the theta divisor Θ s def = Θ X s , there exists an abelian subvariety K i,s of J 1,snc,s of codimension ≤ 1 that stabilizes Θ i,s . (We may assume that K i,s is the maximal abelian subvariety with this property.) The proof of this fact is rather lengthy, but it essentially follows from the generalized Anderson-Indik theorem and Hrushovski's theorem, together with several algebraic-geometric arguments. Now, to prove (0.3), we may replace X by a suitable (finite,étale) covering (see (7.3)). Thus, by (0.8)(ii), we may assume that X is not almost elliptic, and, by (0.8)(i) and (0.7), we may assume d X ≥ 5. Now, by (0.8)(i), Raynaud's theory of theta divisors (see Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:09:02 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:09:02 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf FINITENESS OF ISOMORPHISM CLASSES 683that J 1,snc,s above is not isogenous to a product of elliptic curves. With this assumption, we do not need any inputs from § §2-4. Moreover, under this assumption, we even obtain a stronger result (see (6.5)). In §7, we treat the general case by using the results of § §2-4 fully. Finally, in §8, we present s...
The pro-Galois representation attached to the arithmetic fundamental group of a curve influences heavily the arithmetic of its branched ' -covers.' In many cases, the -power torsion on the Jacobian of such a cover is fixed by the kernel of this representation, giving explicit information about this kernel. Motivated by the relative scarcity of interesting examples for -covers of the projective line minus three points, the authors formulate a conjecture to quantify this scarcity. A proof for certain genus one cases is given, and an exact set of curves satisfying the required arithmetic conditions in the base case is determined.
Let X be an algebraic curve of genus g, n -punctured, defined over a number field K . Then, the profinite or the pro- l completion of the topological fundamental group of X admits two actions: the action of the profinite completion of the mapping class group of the orientable surface of topological type ( g, n ) and the action of the absolute Galois group of K . This paper compares these two. In the profinite case, it is shown that the intersection of the images of these two actions is trivial if X is affine and its fundamental group is nonabelian. On the contrary, in the pro- l case, there are many curves such that the image of the Galois action contains the image of the mapping-class-group action. It is proved that the set of points corresponding to such curves is dense in the moduli space of ( g, n )-curves.
Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p ≥ 0, let f : Y → X be a smooth proper morphism and x a geometric point on X. We prove that the tensor invariants of bounded length ≤ d of π1(X, x) acting on theétale cohomology groups H * (Yx, F ℓ ) are the reduction modulo-ℓ of those of π1(X, x) acting on H * (Yx, Z ℓ ) for ℓ greater than a constant depending only on f : Y → X, d. We apply this result to show that the geometric variant with F ℓ -coefficients of the Grothendieck-Serre semisimplicity conjecture -namely that π1(X, x) acts semisimply on H * (Yx, F ℓ ) for ℓ ≫ 0 -is equivalent to the condition that the image of π1(X, x) acting on H * (Yx, Q ℓ ) is 'almost maximal' (in a precise sense; what we call 'almost hyperspecial') with respect to the group of Q ℓ -points of its Zariski closure. Ultimately, we prove the geometric variant with F ℓ -coefficients of the Grothendieck-Serre semisimplicity conjecture.
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