We give a new way to study recursive towers of curves over a finite field, definedà la Elkies from a bottom curve X and a correspondence Γ on X. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph, Perron-Frobenius theory and considerations on the class of Γ in NS(X × X) lead to the fact that, under some mild assumption, a recursive tower can have in some sense only a restricted asymptotic quality. Results are applied to the Bezerra-Garcia-Stichtenoth tower along the paper for illustration.
We study and compute an infinite family of Hurwitz spaces parameterizing covers of P 1 C branched at four points and deduce explicit regular S n and A n -extensions over Q(T ) with totally real fibers.Introduction.
Using a Hurwitz space computation, we determine the canonical model of the cover of Shimura curves X 0 (2) → X (1) associated to the quaternion algebra over the cubic field of discriminant 13 2 , which is ramified at exactly two real places and unramified at finite places. Then, we list the coordinates of some rational CM points on X (1).
IntroductionIn his classic "Shimura curve computations" [Elk98], Elkies considers Shimura curves from a computational point of view. He explicitly determines covers between Shimura curves associated to the same quaternion algebra and also coordinates of some CM points of those curves. His approach relies mostly on the uniformization of these curves by a quotient of the hyperbolic plane. Several mathematicians have investigated this kind of computational challenge [Voi06,BT07,GR04]. Elkies himself published a second article on the area [Elk06], where he computes the canonical models of some Shimura curves associated with quaternion algebras whose center is one of the cubic cyclic totally real fields of discriminant 49 = 7 2 or 81 = 9 2 and ramified at no finite place and exactly two of the three real places of the center. He ends his article by describing the cover of Shimura curves X 0 (2) → X (1) (see Section 1 for definitions) for the cubic field K of discriminant 13 2 . More precisely, he shows that the curve X (1) has genus zero and that the cover X 0 (2) → X (1) is a degree 9 cover with a certain ramification behavior. But he does not compute equations for the canonical model of this cover. In this work, we compute such a model explicitly.We also illustrate the utility of the computation of Hurwitz spaces. A great tool to compute an explicit model of a cover of P 1 C with given ramification data is to consider the whole family of covers
We provide an infinite sequence of upper bounds for the number of rational points of absolutely irreducible smooth projective curves
X
X
over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many
n
n
-th order Weil bounds, and ending asymptotically at the Drinfeld-Vlăduţ bound. We relate this set of bounds to those of Oesterlé, proving that these are inverse functions in some sense. We explain how the Riemann hypothesis for the curve
X
X
can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface
X
×
X
X\times X
together with the general theory of symmetric Toeplitz matrices. We also give some interpretation for the defect of asymptotically exact towers.
This is achieved by pushing further the classical Weil proof in terms of euclidean relationships between classes in the euclidean part
F
X
\mathcal {F}_X
of the numerical group
Num
(
X
×
X
)
\operatorname {Num}(X\times X)
generated by classes of graphs of iterations of the Frobenius morphism. The noteworthy Toeplitz shape of their intersection matrix takes a central place by implying a very strong cyclic structure on
F
X
\mathcal {F}_X
.
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