## Computational Aspects of Curves of Genus at Least 2 (1996)

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Venue: | Algorithmic number theory. 5th international symposium. ANTS-II |

Citations: | 14 - 3 self |

### BibTeX

@INPROCEEDINGS{Poonen96computationalaspects,

author = {Bjorn Poonen},

title = {Computational Aspects of Curves of Genus at Least 2},

booktitle = {Algorithmic number theory. 5th international symposium. ANTS-II},

year = {1996},

pages = {283--306},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...

### Citations

389 | Modular elliptic curves and Fermat’s last theorem
- Wiles
- 1995
(Show Context)
Citation Context ...allum [75] has used this "method of Chabauty and Coleman" to prove the second case of Fermat's Last Theorem for regular primes. Although this particular application is superseded by the work=-= of Wiles [116]-=- and Taylor-Wiles [111], and preceded by the work of Kummer, who proved Fermat's Last Theorem in its entirety for regular primes, McCallum's work still serves as evidence of the power of the method. W... |

238 |
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern
- Faltings
- 1983
(Show Context)
Citation Context ...Brumer [14]. Assuming the Birch and Swinnerton-Dyer conjecture, these should be the same as the "algebraic" Mordell-Weil ranks. 7. Provably finding all rational points on a curve By Faltings=-=' Theorem [36]-=- (originally the Mordell Conjecture), if a curve over a number field k has genus at least 2, then it has only finitely many k-rational points. Unfortunately the proof is ineffective: it does not provi... |

238 | Ring-theoretic properties of certain Hecke algebras
- Taylor, Wiles
- 1995
(Show Context)
Citation Context ...s "method of Chabauty and Coleman" to prove the second case of Fermat's Last Theorem for regular primes. Although this particular application is superseded by the work of Wiles [116] and Tay=-=lor-Wiles [111]-=-, and preceded by the work of Kummer, who proved Fermat's Last Theorem in its entirety for regular primes, McCallum's work still serves as evidence of the power of the method. When the method of Chaba... |

231 | Advanced Topics in the Arithmetic of Elliptic Curves - Silverman - 1994 |

223 |
Algorithms for modular elliptic curves
- Cremona
- 1992
(Show Context)
Citation Context ...ture projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at =-=[29]-=-. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actu... |

201 |
A subexponential algorithm for discrete logarithms over all finite fields
- Adleman, DeMarrais
- 1993
(Show Context)
Citation Context ...f zeta functions of curves over finite fields.) For concreteness, suppose X is hyperelliptic of genus g = 100 over F 3 , given by an equation y 2 = f(x) with f(x) 2 F 3 [x] of degree 202. It is known =-=[5]-=- that one can compute the number of F 3 -points on the Jacobians of such curves in subexponential time (and even determine the group structure, and solve the discrete logarithm problem), but computing... |

170 |
Elliptic curves over finite fields and the computation of square roots mod p
- Schoof
- 1985
(Show Context)
Citation Context ...ms 2 and 3 above by computing #X(F q m ) in this way for 1smsg. We will refer to this as the naive method. But better techniques are available, at least in theory, if q is large compared to g. Schoof =-=[103]-=- gave a polynomial-time algorithm for computing #X(F q ) where X is an elliptic curve given by a Weierstrass equation in characteristic not equal to 2 or 3. (As usual, polynomial time means polynomial... |

155 |
Computing in Jacobian of a Hyperelliptic Curve,” in
- Cantor
- 1987
(Show Context)
Citation Context ...will be associated with a unique divisor D, some P will have infinitely many pre-images. This problem can sometimes be circumvented by adding additional conditions on D to make it unique. For example =-=[16]-=-, if X is y 2 = f(x) where f is a separable polynomial of degree 2g + 1, then every point on J is represented by a divisor of the form P 1 + P 2 + \Delta \Delta \Delta + P r \Gamma r \Delta 1 where P ... |

152 |
Modular curves and the Eisenstein ideal
- Mazur
- 1977
(Show Context)
Citation Context ...s of Jacobians of non-hyperelliptic curves, except for special curves whose Jacobians have a large endomorphism ring, such as Fermat quotients (see [74] and [55], for example) and modular curves (see =-=[72], for ex-=-ample). For some computations of “analytic ranks” of certain quotients of J0(N), see Brumer [14]. Assuming the Birch and Swinnerton-Dyer conjecture, these should be the same as the “algebraic”... |

145 |
Hyperelliptic cryptosystems
- Koblitz
- 1989
(Show Context)
Citation Context ...in Adleman and Huang's proof that the primes are recognizable in random polynomial time [3]. Finally, Jacobians of hyperelliptic curves over finite fields have been suggested for use in cryptosystems =-=[56]-=-. The security of such systems is dependent on the alleged difficulty of solving the discrete logarithm problem in these algebraic groups. Date: April 10, 1996. This is an extended abstract for an inv... |

111 | On the arithmetic of abelian varieties - Milne - 1972 |

102 |
pour la torsion des courbes elliptiques sur les corps de nombres
- Merel
(Show Context)
Citation Context ... isomorphic to Z=2Z\Theta Z=2NZwith Ns4. The uniform boundedness of the torsion has been generalized to number fields by work of Manin [71], Kamienny and Mazur [52], Abramovich [1], and finally Merel =-=[76]-=-. It is not known whether there is a uniform bound on the size of the torsion subgroup of an abelian variety of fixed dimension gs2 over a fixed number field In fact, there is no bound known even for ... |

94 |
Prolegomena to a middlebrow arithmetic of curves of genus 2
- Cassels, Flynn
- 1996
(Show Context)
Citation Context ..., MAGMA, SIMATH, apecs, and the "Elliptic Curve Calculator." As evidence of the growth of the literature, we note that the first book devoted to the explicit study of genus 2 curves has just=-= appeared [22]-=-. Applications requiring computations with curves of genus at least 2 have existed for well over a century. The oldest (but which has also acquired new relevance since the advent of symbolic integrati... |

83 | On the equations defining abelian varieties - Mumford - 1966 |

52 |
Uniformity of rational points
- Caporaso, Harris, et al.
- 1997
(Show Context)
Citation Context ...light of Faltings' Theorem, it is natural to ask whether the number of krational points on a genus g curve over a number field k can be bounded solely in terms of k and g. Caporaso, Harris, and Mazur =-=[18]-=- have shown that this would follow from some very general conjectures of Lang on rational points on varieties of general type. Abramovich [2] showed more: that the bound could be made uniform for curv... |

50 |
Construction de courbes de genre 2 à partir de leurs modules
- Mestre
- 1991
(Show Context)
Citation Context ..., there are genus 2 curves over Q which are isomorphic to all of their Galois conjugates, but which are not isomorphic to curves defined by polynomial equations over Q. See the end of [105], and also =-=[82]-=-. 4 BJORN POONEN is handed a genus 1 curve and a rational point P , one can find a Weierstrass model simply by computing L(2P ) and L(3P ). If handed a genus 2 curve, compute any canonical divisor K, ... |

50 |
Frobenius maps of abelian varieties and finding roots of unity in finite fields
- Pila
- 1990
(Show Context)
Citation Context ...ntations have been written by Lercier and Morain [66], [67]: they have computed the number of points on elliptic curves over fields of prime order p = 10 499 + 153 and 2-power order q = 2 1301 . Pila =-=[98]-=- gave a theoretical generalization of Schoof's algorithm to curves of higher genus. He proved that for a curve X over F q of any genus, all three problems above can be solved in time O((log q) \Delta ... |

45 |
Primality testing and abelian varieties over finite fields
- Adleman, Huang
- 1992
(Show Context)
Citation Context ...having many points [46], [113]. Also, algorithmic aspects of Jacobians of genus 2 curves play an important role in Adleman and Huang's proof that the primes are recognizable in random polynomial time =-=[3]-=-. Finally, Jacobians of hyperelliptic curves over finite fields have been suggested for use in cryptosystems [56]. The security of such systems is dependent on the alleged difficulty of solving the di... |

44 |
explicites et minorations de conducteurs de variétés algébriques
- Mestre
- 1986
(Show Context)
Citation Context ...sts for elliptic curves in [6] and [29]. 11 10 If a curve has good reduction outside 2, then so does its Jacobian. Thus the list in question should at least contain the 428 curves in [108]. 11 Mestre =-=[79]-=- proves that the conductor N of an g-dimensional abelian variety satisfies N ? (10:32) g , assuming standard conjectures about the L-series. Thus one expects the minimal conductor for genus 2 curves t... |

42 |
les points rationnels des courbes algébriques de genre supérieur à l’unité, Comptes Rendus Hebdomadaires des Séances de l’Acad. des Sci., Paris 212
- Chabauty, Sur
- 1941
(Show Context)
Citation Context ...oints on any given curve. Nevertheless, it is sometimes possible in practice to list all the rational points on a curve by using an idea of Chabauty that predates Faltings' work by 40 years! Chabauty =-=[23]-=- proved that if the Mordell-Weil rank of a curve over a number field k is less than the genus, then the curve has finitely many k-rational points. In order to sketch his idea, let us restrict to the c... |

36 |
2-descent on the Jacobians of Hyperelliptic Curves
- Schaefer
- 1995
(Show Context)
Citation Context ... involved in that it required explicit equations for homogeneous spaces of the Jacobian. Cassels' descent was made explicit and was generalized to hyperelliptic curves over Q of any genus by Schaefer =-=[102]-=- for the odd degree case, and recently by Flynn, Schaefer, and the author [45] for the general even degree case. For concreteness, assume X is a curve y 2 = f(x) where deg f(x) = 5, and J is its Jacob... |

34 |
Moyenne arithmético-géométrique et périodes de courbes de genre 1 et 2
- Bost, Mestre
- 1988
(Show Context)
Citation Context ... be computed using the arithmetic-geometric mean iteration, which amounts to iteratively replacing the curve by a 2-isogenous curve. A generalization to genus 2 was developed by Richelot in 1836. See =-=[10]-=- for a modern treatment. Analytic methods might prove useful in certain situations, for example determining the degrees of possible isogenies between Jacobians of genus 2 curves, but on the other hand... |

34 | Counting the Number of Points of on Elliptic Curves over Finite Fields: Strategies and Performances
- Lercier, Morain
- 1995
(Show Context)
Citation Context ...that made the algorithm computationally viable, and Couveignes [28] developed a practical version for the case of small characterstic. Powerful implementations have been written by Lercier and Morain =-=[66]-=-, [67]: they have computed the number of points on elliptic curves over fields of prime order p = 10 499 + 153 and 2-power order q = 2 1301 . Pila [98] gave a theoretical generalization of Schoof's al... |

32 |
A flexible method for applying Chabauty’s theorem, Compositio Mathematica 105
- Flynn
- 1997
(Show Context)
Citation Context ...ves of the form Dy 2 = x 5 \Gamma x with D 2 Q . 10 BJORN POONEN will actually be able to exhibit this many rational points on X, and then one will know that all rational points have been found. (See =-=[44], [45] and-=- [99] for examples in which this refinement of the method has had success.) McCallum [75] has used this "method of Chabauty and Coleman" to prove the second case of Fermat's Last Theorem for... |

32 | Cycles of Quadratic Polynomials and Rational Points on a Genus 2
- Flynn, Poonen, et al.
- 1997
(Show Context)
Citation Context ...e a function in L(3D) outside the span of f1; x; x 2 ; x 3 g. This yields a model y 2 = f(x) with f(x) of degree 5 or 6. In practice, ad hoc methods for finding nice models can be successful too! See =-=[45]-=- for an example. 4. Computing in the Jacobian of a curve There are at least three different ways of doing computations in the Jacobian J of a curve X of genus gs2. One way is to use the description of... |

32 |
Modular curves and the Eisenstein ideal, Inst
- Mazur
- 1977
(Show Context)
Citation Context ...s of Jacobians of non-hyperelliptic curves, except for special curves whose Jacobians have a large endomorphism ring, such as Fermat quotients (see [74] and [55], for example) and modular curves (see =-=[72], for example).-=- For some computations of "analytic ranks" of certain quotients of J 0 (N ), see Brumer [14]. Assuming the Birch and Swinnerton-Dyer conjecture, these should be the same as the "algebra... |

28 |
On the analogue of the division polynomials for hyperelliptic curves
- Cantor
- 1994
(Show Context)
Citation Context ...rves of the form y 2 = f(x) where f(x) is a separable polynomial of degree 2g + 1 over a field of characteristic not 2. His algorithm requires only O(g 2 log g) field operations to add two points. In =-=[17]-=-, he gives explicit closed form expressions for the multiplication-by-n map on the Jacobian of such a curve, and obtains recurrence relations for calculating the analogues of the division polynomials.... |

28 |
Effective Chabauty
- Coleman
- 1985
(Show Context)
Citation Context ... J(Q). Their intersection is 0-dimensional and in fact finite, and X(Q) maps into this finite set. (This can also be rephrased in terms of the formal group or in terms of p-adic integration.) Coleman =-=[26]-=- was the first to realize that one could give effective bounds for the size of this finite intersection. Using this idea, he was able to show, for example, that if X is a genus g curve over Q with goo... |

28 |
On the Integration of Algebraic Functions
- Davenport
- 1981
(Show Context)
Citation Context ...m of deciding whether the integral of an algebraic function is elementary can be reduced to the problem of deciding whether divisors on algebraic curves represent torsion points on the Jacobian. (See =-=[30]-=- for a detailed discussion.) More recently, the ability to deal with curves of large genus explicitly has had applications in coding theory: to construct efficient algebraic-geometric codes, one needs... |

23 |
Kuyk (eds.), Modular functions of one variable
- Birch, W
- 1975
(Show Context)
Citation Context ...r still, given a genus 2 curve over Q, list all others which have an isogenous Jacobian. ffl Assemble a list of genus 2 curves over Q of small conductor, analogous to the lists for elliptic curves in =-=[6]-=- and [29]. 11 10 If a curve has good reduction outside 2, then so does its Jacobian. Thus the list in question should at least contain the 428 curves in [108]. 11 Mestre [79] proves that the conductor... |

23 |
Computing in the Jacobian of a plane algebraic curve
- Volcheck
(Show Context)
Citation Context ...k. Finally, Volcheck in his thesis described an algorithm, based on some 19th-century methods of Brill and Noether, that solved the problem without assuming the rationality of the singularities. (See =-=[114]-=-, [115].) As alluded to at the end of the last section, a solution to the Riemann-Roch problem can be useful for finding low-degree models of curves. For instance, if one 1 We should warn that not eve... |

20 |
Counting rational points on curves and abelian varieties over nite elds
- Adleman, Huang
(Show Context)
Citation Context ... gave a randomized algorithm in which the exponent \Delta is at worst polynomial in deg f , at least for the case in which the curve has only ordinary multiple points. Very recently Adleman and Huang =-=[4]-=- have given a deterministic algorithm in which \Delta is polynomial in g and N . (But note that while g is at worst polynomial in deg f , the dimension N of the projective space N in which the Jacobia... |

20 |
Galois properties of torsion points on abelian varieties
- Katz
- 1981
(Show Context)
Citation Context ...ar if K = Q and p ? 2), then the entire torsion subgroup injects. By calculating the size of J p (k p ) for various p, one can get an upper bound on the size of the torsion subgroup of J(K). (But see =-=[53]-=- for the limitations of this method.) In practice, there will usually be plenty of small primes of good reduction, so if the genus is reasonably small, the naive method of computing points is sufficie... |

18 |
Construction of rational functions on a curve
- Coates
- 1970
(Show Context)
Citation Context ...s a canonical divisor on X. The Riemann-Roch theorem states that `(D) = deg D + 1 \Gamma g + `(K \Gamma D): The Riemann-Roch problem is to construct explicitly a basis for L(D), given X and D. Coates =-=[25]-=- proved that for curves over algebraic number fields, bases over Q could be effectively constructed. (He needed this for his work with Baker on effective bounds for integer points on elliptic curves [... |

18 | The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field
- Flynn
- 1990
(Show Context)
Citation Context ...series giving the formal group law in terms of two chosen local parameters at the origin on J , have all been given, for y 2 = (quintic) by Grant [48] and for the general case y 2 = (sextic) by Flynn =-=[37]-=-, [40], where the quintic or sextic has indeterminate coefficients. 2 Flynn's formulas are available via anonymous ftp at ftp.liv.ac.uk in the directory e ftp/pub/genus2. The main problem with this ap... |

18 |
Rational torsion of prime order in elliptic curves over number fields, Astérisque (à paraître
- Kamienny, Mazur
(Show Context)
Citation Context ...somorphic to Z=NZwith Ns10 or N = 12, or isomorphic to Z=2Z\Theta Z=2NZwith Ns4. The uniform boundedness of the torsion has been generalized to number fields by work of Manin [71], Kamienny and Mazur =-=[52]-=-, Abramovich [1], and finally Merel [76]. It is not known whether there is a uniform bound on the size of the torsion subgroup of an abelian variety of fixed dimension gs2 over a fixed number field In... |

17 |
Computing the Mordell-Weil rank of Jacobians of curves of genus 2
- Gordon, Grant
- 1993
(Show Context)
Citation Context ... is finite. Here we will describe the generalization of one of these methods, 2-descent, to the case of hyperelliptic curves. Cassels outlined an approach for genus 2 curves in [20]. Gordon and Grant =-=[47]-=- carried this out for some curves, but their method worked only in the very special case where all six Weierstrass points were rational, and the method was quite involved in that it required explicit ... |

17 |
On the method of Coleman and
- McCallum
- 1994
(Show Context)
Citation Context ...his many rational points on X, and then one will know that all rational points have been found. (See [44], [45] and [99] for examples in which this refinement of the method has had success.) McCallum =-=[75] has used -=-this "method of Chabauty and Coleman" to prove the second case of Fermat's Last Theorem for regular primes. Although this particular application is superseded by the work of Wiles [116] and ... |

16 |
Integer points on curves of genus 1
- Baker, Coates
- 1970
(Show Context)
Citation Context ...] proved that for curves over algebraic number fields, bases over Q could be effectively constructed. (He needed this for his work with Baker on effective bounds for integer points on elliptic curves =-=[7]-=-.) Much more recently, Huang and Ierardi [50] proved that the problem could be solved over the ground field k, and in polynomial time, for plane curves whose singularities are all defined over k. Fina... |

16 |
Quelques calculs en théorie des nombres. Thèse, Université de Bordeaux I
- Couveignes
- 1994
(Show Context)
Citation Context ...e; the naive method, in contrast, requires time slightly worse than linear in q.) Subsequently, Atkin and Elkies introduced improvements that made the algorithm computationally viable, and Couveignes =-=[28]-=- developed a practical version for the case of small characterstic. Powerful implementations have been written by Lercier and Morain [66], [67]: they have computed the number of points on elliptic cur... |

15 |
Heegner point computations, Algorithmic number theory
- Elkies
- 1994
(Show Context)
Citation Context ...the intersection exactly, and this leads to an improved upper bound for #X(Q). With luck, one 5 This actually happens in genus 1: for instance, for the rank 1 elliptic curve 1063y 2 = x 3 \Gamma x of =-=[33]-=-, the x-coordinate of a generator of the Mordell-Weil group modulo torsion is X 2 =1063 where X = 11091863741829769675047021635712281767382339667434645 317342657544772180735207977320900012522807936777... |

14 | et discriminant minimal de courbes de genre 2 - Liu - 1994 |

14 |
The p-torsion of elliptic curves is uniformly bounded
- Manin
- 1969
(Show Context)
Citation Context ...onal points of X can be found in the (finite) pre-image of the rational points on that elliptic curve. This is a trivial instance of a general method of Dem'janenko [32], further generalized by Manin =-=[71]-=- 6 : if X is a curve over a number field k, if A is a k-simple abelian variety such that A m occurs in the decomposition of the Jacobian of X up to isogeny over k, and if m ? rankA(k) rank End k A ; t... |

13 |
Un théorème d’arithmétique sur les courbes algébriques
- Chevalley
- 1932
(Show Context)
Citation Context ...ctive. See [106] for some explicit applications of this method. One can also attempt to use unramified covers of X: if Y is an unramified cover of X, then according to a theorem of Chevalley and Weil =-=[24]-=-, there is a certain extension field k 0 such that the pre-images of the rational points on X are contained in Y (k 0 ). Although Y will have higher genus than X if the genus of X is at least 2 (and i... |

13 |
Rational points on certain elliptic modular curves
- Ogg
- 1973
(Show Context)
Citation Context ...ch is available via anonymous ftp at megrez.math.u-bordeaux.fr in the directory /pub/liu, also computes the odd 7 The existence for ` = 19 and ` = 21 was in fact demonstrated over 20 years ago by Ogg =-=[94]-=-: the 2-dimensional modular Jacobians J 1 (13) and J 1 (18) have torsion subgroups isomorphic to Z=19Zand Z=21Z, respectively. The only other J 1 (N) of dimension 2 is J 1 (16), whose torsion subgroup... |

13 |
The classification of rational preperiodic points of quadratic polynomials over Q: a refined conjecture
- Poonen
- 1998
(Show Context)
Citation Context ...very restricted class of genus 2 curves. Stoll also has implemented a 2-descent for most curves of the form y 2 = x 5 + D. The deg f = 6 algorithm has been successfully used a few times (see [45] and =-=[99]-=-), but no one has automated it yet. Stoll also written a program that computes lower bounds on the rank of J(Q) by attempting to find the exact rank of a subgroup generated by a given set of points by... |

13 | Abelian varieties over Q with large endomorphism algebras and their simple components over Q. Modular curves and abelian varieties - Pyle |

12 |
The Mordell-Weil group of curves of genus 2
- Cassels
- 1983
(Show Context)
Citation Context ... Shafarevich-Tate group is finite. Here we will describe the generalization of one of these methods, 2-descent, to the case of hyperelliptic curves. Cassels outlined an approach for genus 2 curves in =-=[20]-=-. Gordon and Grant [47] carried this out for some curves, but their method worked only in the very special case where all six Weierstrass points were rational, and the method was quite involved in tha... |

11 |
Uniformité des points rationnels des courbes algébriques sur les extensions quadratiques et cubiques
- Abramovich
(Show Context)
Citation Context ... injective homomorphism J(Q)=2J(Q) x\GammaT \Gamma! i ker : L =L 2 Norm ! Q =Q 2 j : Schaefer [102] proved that i ker : L =L 2 Norm ! Q =Q 2 j was isomorphic to the Galois cohomology group H 1 (GQ ; J=-=[2]), and tha-=-t under this identification the "(x\GammaT )" map coincided with the usual coboundary map of Galois cohomology. Moreover he demonstrated how to compute the 2-Selmer group of J explicitly as ... |

11 | Descent via isogeny in dimension 2 - Flynn - 1994 |