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Degree 4 Coverings of Elliptic Curves by Genus 2 Curves
Abstract
Genus two curves covering elliptic curves have been the object of study of
many articles. For a fixed degree n the subloci of the moduli space of curves having a degree n elliptic subcover has been computed for
and discussed in detail for n odd; see \cite{Sh1, SV2, Fr, FK}. When
the degree of the cover is even the case in general has been treated in
\cite{PRS}. In this paper we compute the sublocus of of curves
having a degree 4 elliptic subcover.
... On the turn of the new century a couple of papers appeared for the case of genus g = 2; see [14,40]. For more details on similar topics the reader can check further [4,21,23,24,29,30,34,39]. ...
... These invariants satisfy Eqs. (30). Hence, the curve is in the (Z 2 × Z 4 )-locus. ...
In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field k and the intersection of the moduli space \M_3^b of such curves with the hyperelliptic moduli \H_3. Such intersection is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the -invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization
of . We also compute all possible subloci of curves for all possible automorphism group G. Moreover, for every rational moduli point \p \in \S, such that | \Aut (\p) | > 4, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the Gl(2,k)-invariants.
... Lately the space L 5 has been studied in detail in [10]. The case of even degree has been less studied even though there have been some attempts lately to compute some of the cases for n = 4; see [4]. In this survey we study the genus 2 curves with (n, n)-split Jacobian for small n. ...
In this survey we study the genus 2 curves with -split Jacobian for even n.
... This goes back to Legendre and Jacobi for n = 2, and to Hermite, Goursat and Bolza among others for n = 3 and 4. This topic has been revisited more recently by Kuhn for n = 3 (see [Kuh88]), and by Shaska and his collaborators for 2 ≤ n ≤ 5 (see [Sha04], [MSV09], and [SWWW08]). The connection of this problem to split Jacobians is that for such C and E, the Jacobian Jac(C) is isogenous to the product of elliptic curves E × E ′ , where E ′ = Jac(C)/φ * (E). ...
The main result of [FG20] classifies the 92 geometric endomorphism algebras of geometrically split abelian surfaces defined over Q. We show that 54 of them arise as geometric endomorphism algebras of Jacobians of genus 2 curves defined over Q, and that the remaining 38 do not. In particular, we exhibit 38 examples of Qbar-isogeny classes of abelian surfaces defined over Q that do not contain any Jacobian of a genus 2 curve defined over Q, and for each of the 54 algebras that do arise we exhibit a curve realizing them.
... Kuhn [16] revisited this topic, giving a combinatorial description in terms of the branch points of the map φ as well as the hyperelliptic involution ι on C, and adapting some examples to positive characteristic. This approach has led to fairly explicit descriptions of L n for 2 ≤ n ≤ 5 by Shaska and others [23,[26][27][28]. From an arithmetic standpoint, reducible Jacobians for 2 ≤ n ≤ 4 have been studied in [2,3,5]. ...
We compute explicit rational models for some Hilbert modular surfaces
corresponding to square discriminants, by connecting them to moduli spaces of
elliptic K3 surfaces. Since they parametrize decomposable principally polarized
abelian surfaces, they are also moduli spaces for genus-2 curves covering
elliptic curves via a map of fixed degree. We thereby extend classical work of
Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska,
V\"olklein, Magaard and others, producing explicit families of reducible
Jacobians. In particular, we produce a birational model for the moduli space of
pairs (C,E) of a genus 2 curve C and elliptic curve E with a map of degree n
from C to E, as well as the universal family over the base, for 2 <= n <= 11.
We also analyze the resulting models from the point of view of arithmetic
geometry, and produce some interesting curves on them.
... Lately the space L 5 has been studied in detail in [10]. The case of even degree has been less studied even though there have been some attempts lately to compute some of the cases for n = 4; see [4]. In this survey we study the genus 2 curves with (n, n)-split Jacobian for small n. ...
In this survey we study the genus 2 curves with -split Jacobian
for even n.
... The cases of odd degree have been studied for deg π i = 3, 5; see [12, 18] respectively. For even degree the only case which has been studied computationally is degree 4; see [17] and [6]. For a detailed study of the case of genus g = 2; see [4]. ...
We investigate the decomposition of Jacobians of superelliptic curves based
on their automorphisms. For curve with equation we provide an
necessary and sufficient condition in terms of m and n for the
decomposition of the Jacobian induced by the automorphisms of the curve.
Moreover, we generalize a construction in \cite{Ya} of a family of
non-hyperelliptic curves \X_{r,s} and determine arithmetic conditions on r
and s that the Jacobians \J (\X_{r, s}) decomposes.
... Such curves have been of much interest lately because of their use in many theoretical and applicative situations. The first part of the paper is based on several papers on the topic of genus two curves with split Jacobians; see [1,[3][4][5][6][7][8][9][11][12][13][14][16][17][18][19][20][21] among others. ...
We study the singular locus on the algebraic surface of genus 2 curves
with a -split Jacobian. Such surface was computed by Shaska in
\cite{deg3} for n=3, and Shaska at al. in \cite{deg5} for n=5. We show that
the singular locus for n=2 is exactly th locus of the curves of automorphism
group or . For n=3 we use a birational parametrization of the
surface discovered in \cite{deg3} to show that the singular locus is a
0-dimensional subvariety consisting exactly of three genus 2 curves (up to
isomorphism) which have automorphism group or . We further show that
the birational parametrization used in would work for all if
is a rational surface.
We build a database of genus 2 curves defined over \Q which contains all curves with minimal absolute height \h \leq 5, all curves with moduli height \mH \leq 20, and all curves with extra automorphisms in standard form defined over \Q with height \h \leq 101. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants.
The distribution of rational points in the moduli space \M_2 for which the field of moduli is a field of definition is discussed and some open problems are presented.
Për mendimin tim, Kalkulusi duhet bërë në shkollën e mesme. Të paktën për ata nxënës që duan të vazhdojnë në degët e shkencave ose inxhinjerive. Leksionet e mëposhtme kanë plot gabime nëpër to sepse nuk i kanë kaluar provës në klasë nga qindra nxënës dhe mësues. Megjithate, për mendimin tim, janë të vetmet në Shqip që synojnë të pregatisin nxënësit me atë bagazh matematike që konsiderohet standart për studentët e shkencave dhe inxhinjerive. Ata mësues që duan ti përdorin këto leksione janë të mirëpritur ta bëjnë këtë dhe kur gjejnë gabime nëpër to ju lutem të më thonë që ti korrigjojmë.
Për mendimin tim, Kalkulusi 3 duhet bërë në shkollën e mesme. Të paktën për ata nxënës që duan të vazhdojnë në degët e shkencave ose inxhinjerive.
Leksionet e mëposhtme kanë plot gabime nëpër to sepse nuk i kanë kaluar provës në klasë nga qindra nxënës dhe mësues. Megjithate, për mendimin tim, janë të vetmet në Shqip që synojnë të pregatisin nxënësit me atë bagazh matematike që konsiderohet standart për studentët e shkencave dhe inxhinjerive.
Ata mësues që duan ti përdorin këto leksione janë të mirëpritur ta bëjnë këtë dhe kur gjejnë gabime nëpër to ju lutem të më thonë që ti korrigjojmë.
Kurbat algjebrike janë nje nga objektet me të rëndësishme të matematikës. Disa nga problemet me tradicionale të matematikës janë pikërisht mbi kurbat algjebrike. Gjithashtu, me zhvillimin e informatikës dhe mënyrave te reja të llogaritjeve kurbat algjebrike kanë marrë nje rëndësi të vecantë. Ato tani përdoren gjerësisht në kryptografi, teorinë e kodeve dhe mjaft fusha të tjera. Në këtë libër unë jam përpjekur të jap nje trajtim të shkurtër dhe konciz të kasaj dege të gjeometrisë algjebrike.
This paper is the first version of a project of classifying all superelliptic
curves of genus according to their automorphism group. We determine
the parametric equations in each family, the corresponding signature of the
group, the dimension of the family, and the inclussion among such families. At
a later stage it will be determined the decomposition of the Jacobians and each
locus in the moduli spaces of curves.
Ky liber eshte nje hyrje ne algjebren abstrakte per studentët e matematikës. Teksti supozohet të përdoret si tekst bazë puer studentuet e matematikës gjatë një periudhe prej dy semestrash.
This paper is the first version of a project of classifying all su-perelliptic curves of genus g ≤ 48 according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group, the dimension of the family, and the inclussion among such families. At a later stage it will be determined the decomposition of the Jacobians and each locus in the moduli spaces of curves.
Ky libër eshte nje hyrje ne algjebren abstrakte per studentet e degeve te matematikes. Mund te perdoret me sukses edhe per studentet e degeve te tjera si shkencave kompjuterike, inxhinjerise elektrike, teorise se komunikacineve, etj.
Algjebra abstrakte eshte jo vetem nje nga deget me te bukura dhe me klasike te matematikes, por edhe nje nga deget me aktive te saj. Sot algjebra abstrakte ka aplikime ne fusha te ndryshme te shkences, industrise, dhe ekonomise, si per shembull në teorine e komunikimeve dixhitale, teorine e kodeve, kriptografi, sistemet financiare, robotike, pervec aplikimeve te fiziken teorike, kimi, biomatematike, etj.
Libri është ndarë në katër pjesë, në teorinë e grupeve, unazave, moduleve, dhe teorinë e fushave.
Let G be a finite group, and . We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g=3 (including equations for the corresponding curves), and for we classify those loci corresponding to "large" G.
Over the last 15 years important results have been achieved in the field of Hilbert Modular Varieties. Though the main emphasis of this book is on the geometry of Hilbert modular surfaces, both geometric and arithmetic aspects are treated. An abundance of examples - in fact a whole chapter - completes this competent presentation of the subject. This Ergebnisbericht will soon become an indispensible tool for graduate students and researchers in this field.
Basic invariants of binary forms over ℂ up to degree 6 (and lower degrees) were constructed by Clebsch and Bolza in the 19-th century using complicated symbolic calculations. Igusa extended this to algebraically closed fields of any characteristic using difficult techniques of algebraic geometry. In this paper a simple proof is supplied that works in characteristic p > 5 and uses some concepts of invariant theory developed by Hilbert (in characteristic 0) and Mumford, Haboush et al. in positive characteristic. Further the analogue for pairs of binary cubics is also treated.
As an illustration of the previous chapters we now give examples. Reading this chapter will resemble a visit to a zoological garden: each of the surfaces presented here is a world in itself that can be explored as far as one wishes, either for its own peculiarities or to test general principles.
We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and a map from X to an elliptic curve, then X has split jacobian.It is not true that a complement to E in the jacobian of X is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve and algebraic , and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics 0 and 3.
In this survey we study the genus 2 curves with -split Jacobian
for even n.
The purpose of the following paper is to show how one can relate elliptic curves E with covering curves C of genus 2 and arithmetical properties of E and C in the case that the ground field is a global field.
The system of binary forms is finite. I proved this theorem and Mr. Hilbert extended it to forms in n variables. In this note, a theorem is given which implies the others.
A genus 2 curve C has an elliptic subcover if there exists a degree n maximal covering to an elliptic curve E. Degree n elliptic subcovers occur in pairs . The Jacobian of C is isogenous of degree to the product . We say that is -split. The locus of C, denoted by \L_n, is an algebraic subvariety of the moduli space \M_2. The space \L_2 was studied in Shaska/V\"olklein and Gaudry/Schost. The space \L_3 was studied in Shaska (2004) were an algebraic
description was given as sublocus of \M_2. In this survey we give a brief description of the spaces \L_n for a general n and then focus on small n. We describe some of the computational details
which were skipped in Shaska/V\"olklein and Shaska (2004). Further we
explicitly describe the relation between the elliptic subcovers E and .
We have implemented most of these relations in computer programs which check
easily whether a genus 2 curve has or split Jacobian. In each
case the elliptic subcovers can be explicitly computed.
We continue our study of genus 2 curves C that admit a cover to
a genus 1 curve E of prime degree n. These curves C form an irreducible
2-dimensional subvariety \L_n of the moduli space \M_2 of genus 2 curves.
Here we study the case n=5. This extends earlier work for degree 2 and 3,
aimed at illuminating the theory for general n.
We compute a normal form for the curves in the locus \L_5 and its three
distinguished subloci. Further, we compute the equation of the elliptic
subcover in all cases, give a birational parametrization of the subloci of
\L_5 as subvarieties of \M_2 and classify all curves in these loci which
have extra automorphisms.
We consider the moduli spaceS
n
of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS
n
is viewed as a subset of the moduli spaceA
2 of principally polarized abelian surfaces. On the other hand the Humbert surfaceH
Δ of invariant Δ is defined as a subvariety ofA
2(C), the set of C-valued points ofA
2. The purpose of this paper is to releaseS
n
withH
Δ.
Let C be a curve of genus 2 and ψ1: C − → E 1 a map of degree n, from C to an elliptic curveE1 , both curves defined over C. This map induces a degree n map φ1:P1 − → P 1 which we call a Frey–Kani covering. We determine all possible ramifications for φ1. If ψ1:C − → E 1 is maximal then there exists a maximal map ψ2: C − → E 2 , of degree n, to some elliptic curveE2 such that there is an isogeny of degree n2from the JacobianJC to E1 × E2. We say thatJC is (n, n)-decomposable. If the degree n is odd the pair (ψ2, E2) is canonically determined. For n = 3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.
Let C be a genus 2 curve defined over k, char(k)=0. If C has a (3,3)-split Jacobian then we show that the automorphism group Aut(C) is isomorphic to one of the following: Z(2), V-4, D-8, or D-12. There axe exactly six C-isomorphism classes of genus two curves C with Aut(c) isomorphic to D8 (resp., D-12) and with (3,3)-split Jacobian. We show that exactly four (resp., three) of these classes with group D-8 (resp., D-12) have representatives defined over Q. We discuss some of these curves in detail and find their rational points.
The group of m-torsion points on an elliptic curve, for a prime number m, forms a two-dimensional vector space. It was sug- gested and proven by Yoshida that under certain conditions the vector decomposition problem (VDP) on a two-dimensional vector space is at least as hard as the computational Die-Hellman problem (CDHP) on a one-dimensional subspace. In this work we show that even though this assessment is true, it applies to the VDP for m-torsion points on an el- liptic curve only if the curve is supersingular. But in that case the CDHP on the one-dimensional subspace has a known sub-exponential solution. Furthermore, we present a family of hyperelliptic curves of genus two that are suitable for the VDP.
Printout. Thesis (Ph. D.)--University of Florida, 2001. Vita. Includes bibliographical references (leaves 69-70).
The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus \L_g of such genus g hyperelliptic curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves \H_g. We discover a birational parametrization of \L_g via dihedral invariants and show how these invariants can be used to determine the field of moduli of points \p \in \L_g. We conjecture that for \p\in \H_g with |\Aut(\p)| > 2 the field of moduli is a field of definition and prove this conjecture for any point \p\in \L_g such that the Klein 4-group is embedded in the reduced automorphism group of \p. Further, for g=3 we show that for every moduli point \p \in \H_3 such that | \Aut (\p) | > 4, the field of moduli is a field of definition and provide a rational model of the curve over its field of moduli.
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be approached from a computational viewpoint.
In this paper we study genus 2 function fields K with degree 3 elliptic
subfields. We show that the number of Aut(K)-classes of such subfields of K is
0,1,2, or 4. Also we compute an equation for the locus of such K in the moduli
space of genus 2 curves.
Genus 2 curves with degree 5 elliptic subcovers
- K Magaard
- T Shaska
- H Völklein
K. Magaard, T. Shaska, H. V¨ olklein, Genus 2 curves with degree 5 elliptic subcovers, Forum Math. (to appear).
Review of Legendre, Théorie des fonctions elliptiques. Troiseme supplém ent. 1832
- C Jacobi
C. Jacobi, Review of Legendre, Théorie des fonctions elliptiques. Troiseme supplém ent.
1832. J. reine angew. Math. 8, 413-417.
Elliptic subfields and automorphisms of genus two fields, Algebra, Arithmetic and Geometry with Applications, pg
- T Shaska
- H Völklein
T. Shaska and H. Völklein, Elliptic subfields and automorphisms of genus two fields,
Algebra, Arithmetic and Geometry with Applications, pg. 687 -707, Springer (2004).
Invariants des quotients de la Jacobienne d’une courbe de genre 2
- P Gaudry
- E Schost
P. Gaudry and E. Schost, Invariants des quotients de la Jacobienne d'une courbe de genre
2, (in press)
Sur les fonctionnes abliennes singulires. I, II, III
- G Humbert
G. Humbert Sur les fonctionnes abliennes singulires. I, II, III. J. Math. Pures Appl. serie 5,
t. V, 233-350 (1899); t. VI, 279-386 (1900);
- A Krazer
A. Krazer, Lehrbuch der Thetafunctionen, Chelsea, New York, 1970.
Genus 2 curves with degree 3 elliptic subcovers
- T Shaska