Specializations of Galois covers of the line

2011 ◽  
Author(s):  
Pierre Dèbes ◽  
Nour Ghazi
Keyword(s):  
Galois Covers

scholarly journals Decomposing Jacobians Via Galois covers

2021 ◽  
pp. 1-23
Author(s):  
Davide Lombardo ◽  
Elisa Lorenzo García ◽  
Christophe Ritzenthaler ◽  
Jeroen Sijsling
Keyword(s):  
Galois Covers

On Galois covers of Hirzebruch surfaces

1996 ◽  
Vol 305 (1) ◽  
pp. 493-539 ◽  
Author(s):  
B. Moishezon ◽  
A. Robb ◽  
M. Teicher
Keyword(s):  
Hirzebruch Surfaces ◽  
Galois Covers

�tale Galois covers of affine smooth curves

1995 ◽  
Vol 120 (1) ◽  
pp. 555-578 ◽  
Author(s):  
Florian Pop
Keyword(s):  
Smooth Curves ◽  
Galois Covers

scholarly journals DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

2019 ◽  
pp. 1-42
Author(s):  
Joachim König ◽  
François Legrand
Keyword(s):  
Hyperelliptic Curves ◽  
Branch Points ◽  
Abc Conjecture ◽  
Galois Cover ◽  
Quadratic Twists ◽  
Large Genus ◽  
Galois Covers ◽  
The One ◽  
Almost All ◽  
Global Principle

We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ , almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$ . We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$ -rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ . We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$ . On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

scholarly journals Limits of canonical forms on towers of Riemann surfaces

2020 ◽  
Vol 2020 (764) ◽  
pp. 287-304
Author(s):  
Hyungryul Baik ◽  
Farbod Shokrieh ◽  
Chenxi Wu
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Type Theorem ◽  
Universal Cover ◽  
Canonical Forms ◽  
Classical Version ◽  
Galois Cover ◽  
Hyperbolic Form ◽  
Galois Covers ◽  
Hyperbolic Riemann Surface

AbstractWe prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on {S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.

Galois Covers of an Arithmetic Surface

1988 ◽  
Vol 110 (5) ◽  
pp. 849 ◽  
Author(s):  
David Harbater
Keyword(s):  
Arithmetic Surface ◽  
Galois Covers

scholarly journals Galois covers of the open p-adic disc

2009 ◽  
Vol 131 (1-2) ◽  
pp. 43-61 ◽  
Author(s):  
Scott Corry
Keyword(s):  
Galois Covers

Towers of GL($r$)-type of modular curves

2019 ◽  
Vol 2019 (754) ◽  
pp. 87-141 ◽  
Author(s):  
Ernst-Ulrich Gekeler
Keyword(s):  
Rational Points ◽  
Galois Groups ◽  
Modular Curves ◽  
Galois Covers

Abstract We construct Galois covers {X^{r,k}(N)} over {{\mathbb{P}}^{1}/{\mathbb{F}}_{q}(T)} with Galois groups close to {{\rm GL}(r,{\mathbb{F}}_{q}[T]/(N))} ( {r\geq 3} ) and rationality and ramification properties similar to those of classical modular curves {X(N)} over {{\mathbb{P}}^{1}/{\mathbb{Q}}} . As application we find plenty of good towers (with \limsup{\frac{\text{number~{}of~{}rational~{}points}}{{\rm genus}}>0} ) of curves over the field {{\mathbb{F}}_{q^{r}}} with {q^{r}} elements.

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