1,398 research outputs found
A note on the stratification by automorphisms of smooth plane curves of genus 6
In this note, we give a so-called representative classification for the
strata by automorphism group of smooth -plane curves of genus ,
where is a fixed separable closure of a field of characteristic
or . We start with a classification already obtained by the
first author and we use standard techniques.
Interestingly, in the way to get these families for the different strata, we
find two remarkable phenomenons that did not appear before. One is the
existence of a non -dimensional final stratum of plane curves. At a first
sight it may sound odd, but we will see that this is a normal situation for
higher degrees and we will give a explanation for it.
We explicitly describe representative families for all strata, except for the
stratum with automorphism group . Here we find the
second difference with the lower genus cases where the previous techniques do
not fully work. Fortunately, we are still able to prove the existence of such
family by applying a version of Luroth's theorem in dimension
Integer sequences that are generalized weights of a linear code
Which integer sequences are sequences of generalized weights of a linear
code? In this paper, we answer this question for linear block codes,
rank-metric codes, and more generally for sum-rank metric codes. We do so under
an existence assumption for MDS and MSRD codes. We also prove that the same
integer sequences appear as sequences of greedy weights of linear block codes,
rank-metric codes, and sum-rank metric codes. Finally, we characterize the
integer sequences which appear as sequences of relative generalized weights
(respectively, relative greedy weights) of linear block codes.Comment: 19 page
Torcimiento de cuárticas planas lisas
En este trabajo se determinan los twists de las cuárticas planas lisas, es decir, de las curvas lisas no hiperelĂpticas de gĂ©nero 3. Se parte de una clasificaciĂłn de Henn de las cuárticas planas y sus grupos de automorfismos y a partir de los conjuntos de cohomologĂa de Galois de estos grupos se determinan los twists.. Estudiar les propietats aritmètiques de les corbes de gènere 3 no hiperel·lĂptiques, amb especial èmfasi en la part computacional sobre l'efecte torçament; cĂ lculs en cohomologia de Galois, acciĂł torçada sobre l'espai invariant de diferencials regulars, etc
Torcimiento de cuárticas planas lisas
En este trabajo se determinan los twists de las cuárticas planas lisas, es decir, de las curvas lisas no hiperelĂpticas de gĂ©nero 3. Se parte de una clasificaciĂłn de Henn de las cuárticas planas y sus grupos de automorfismos y a partir de los conjuntos de cohomologĂa de Galois de estos grupos se determinan los twists.. Estudiar les propietats aritmètiques de les corbes de gènere 3 no hiperel·lĂptiques, amb especial èmfasi en la part computacional sobre l'efecte torçament; cĂ lculs en cohomologia de Galois, acciĂł torçada sobre l'espai invariant de diferencials regulars, etc
Primes dividing invariants of CM Picard curves
We give a bound on the primes dividing the denominators of invariants of
Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in
genus 2 and 3, our bound is based not on bad reduction of curves, but on a very
explicit type of good reduction. This approach simultaneously yields a
simplification of the proof, and much sharper bounds. In fact, unlike all
previous bounds for genus 3, our bound is sharp enough for use in explicit
constructions of Picard curves
Arithmetic properties of non-hyperelliptic genus 3 curves
This thesis explores the explicit computation of twists of curves. We develope an algorithm for computing the twists of a given curve assuming that its automorphism group is known. And in the particular case in which the curve is non-hyperelliptic we show how to compute equations of the twists. The algorithm is based on a correspondence that we establish beetwen the set of twists and the set of solutions of a certain Galois embedding problem. In general is not known how to compute all the solution of a Galois embedding problem. Throughout the thesis we give some ideas of how to solve these problems.
The twists of curves of genus less or equal than 2 are well-known. While the genus 0 and 1 cases go back from long ago, the genus 2 case is due to the work of Cardona and Quer. All the genus 0, 1 or 2 curves are hyperelliptic, however for genus greater than 2 almost all the curves are non-hyperelliptic.
As an application to our algorithm we give a classification with equations of the twists of all plane quartic curves, that is, the non-hyperelliptic genus 3 curves, defined over any number field k. The first step for computing such twists is providing a classification of the plane quartic curves defined over a concrete number field k. The starting point for doing this is Henn classification of plane quartic curves with non-trivial automorphism group over the complex numbers.
An example of the importance of the study of the set of twists of a curve is that it has been proven to be really useful for a better understanding of the behaviour of the Generalize Sato-Tate conjecture, see the work of Fité, Kedlaya and Sutherland. We show a proof of the Sato-Tate conjecture for the twists of the Fermat and Klein quartics as a corollary of a deep result of Johansson, and we compute the Sato-Tate groups and Sato-Tate distributions of them.
Following with the study of the Generalize Sato-Tate conjecture, in the last chapter of this thesis we explore such conjecture for the Fermat hypersurfaces X_{n}^{m}: x_{0}^{m}+...+x_{n+1}^{m} = 0. We explicitly show how to compute the Sato-Tate groups and the Sato-Tate distributions of these Fermat hypersurfaces. We also prove the conjecture over the rational numbers for n=1 and over than the cyclotomic field of mth-roots of the unity if n is greater 1.En esta tesis estudiamos el cálculo explĂcito de twists de curvas. Se desarrolla un algoritmo para calcular los twists de una curva dada asumiendo que su grupo de automorfismos en conocido. Además, en el caso particular en que la curva es no hiperelĂptica se enseña como calcular ecuaciones de los twists. El algoritmo está basado es una correspondencia que establecemos entre el conjunto de twists de la curva y el conjunto de soluciones a un cierto problema de embeding de Galois. Aunque no existe un mĂ©todo general para resolver este tipo de problemas a lo largo de la tesis se exponen algunas ideas para resolver algunos de estos problemas en concreto. Los twists de curvas de gĂ©nero menor o igual que 2 son bien conocidos. Mientras que los casos de gĂ©nero 0 y 1 se conocen desde hace tiempo, el caso de gĂ©nero 2 es más reciente y se debe al trabajo de Cardona y Quer. Todas las curvas de gĂ©nero, 0,1 y 2 son hiperelĂpticas, sin embargo, las curvas de gĂ©nero mayor o igual que 3 son en su mayorĂa no hipèrelĂpticas. Como aplicaciĂłn a nuestro algoritmo damos una clasificaciĂłn con ecuaciones de los twists de todas las cuárticas planas lisas, es decir, de todas las curvas no hiperelĂpticas de gĂ©nero 3, definidas sobre un cuerpo de nĂşmeros k. El primer paso para calcualr estos twists es obtener una clasificaciĂłn de las cuárticas planas lisas definidas sobre un cuerpo de nĂşmeros k arbitrario. El punto de partida para obtener esta clasificaciĂłn es la clasificaciĂłn de Henn de cuárticas planas definidas sobre los nĂşmeros complejos y con grupo de automorfismos no trivial. Un ejemplo de la importancia del estudio de los twists de curvas es que se ha probado que resulta ser de gran utilidad para el mejor entendimiento del carácter de la conjetura de Sato-Tate generalizada, como puede verse en los trabajos de entre otros: FitĂ©, Kedlaya y Sutherland. En la tesis se prueba la conjetura de Sato-Tate para el caso de los twists de las cuárticas de Fermat y de Klein como corolario de un resultado de Johansson, además se calculan los grupos y las distribuciones de Sato-Tate de estos twists. Siguiendo con el estudio de la conjetura generalizada de Sato-Tate, en el Ăşltimo capĂtulo de la tesis se estudia la conjetura para el caso de las hipersuperficies de Fermat: X_{n}^{m}: x_{0}^{m}+...+x_{n+1}^{m} = 0. Se muestra esplĂcitamente como calcular los grupos de Sato-Tate y las correspondientes distribuciones. Además se prueba la conjetura para el caso n=1 sobre el cuerpo de los nĂşmeros racionales y para n mayor que 1 sobre el cuerpo de las raĂces m-Ă©simas de la unidad
Refinements of Katz-Sarnak theory for the number of points on curves over finite fields
This paper goes beyond Katz-Sarnak theory on the distribution of curves over
finite fields according to their number of rational points, theoretically,
experimentally and conjecturally. In particular, we give a formula for the
limits of the moments measuring the asymmetry of this distribution for
(non-hyperelliptic) curves of genus . The experiments point to a
stronger notion of convergence than the one provided by the Katz-Sarnak
framework for all curves of genus . However, for elliptic curves and
for hyperelliptic curves of every genus we prove that this stronger convergence
cannot occur.Comment: 22 pages, 5 figure
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