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Journal articles on the topic 'Genus 2 curves'

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Author: Grafiati

Published: 4 May 2024

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1

Baba, Srinath, and Håkan Granath. "Genus 2 Curves with Quaternionic Multiplication." Canadian Journal of Mathematics 60, no. 4 (August 1, 2008): 734–57. http://dx.doi.org/10.4153/cjm-2008-033-7.

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AbstractWe explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QMcurves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our j-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using j, we construct the fields ofmoduli and definition for somemoduli problems associated to the Atkin–Lehner group actions.

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2

González-Jiménez, Enrique, and Josep González. "Modular curves of genus 2." Mathematics of Computation 72, no. 241 (June 4, 2002): 397–419. http://dx.doi.org/10.1090/s0025-5718-02-01458-8.

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3

Cosset, Romain. "Factorization with genus 2 curves." Mathematics of Computation 79, no. 270 (August 20, 2009): 1191–208. http://dx.doi.org/10.1090/s0025-5718-09-02295-9.

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4

Mourao, Michael. "Extending Elliptic Curve Chabauty to higher genus curves." Manuscripta Mathematica 143, no. 3-4 (April 5, 2013): 355–77. http://dx.doi.org/10.1007/s00229-013-0621-2.

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5

Bröker, Reinier, Everett W. Howe, Kristin E. Lauter, and Peter Stevenhagen. "Genus-2 curves and Jacobians with a given number of points." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 170–97. http://dx.doi.org/10.1112/s1461157014000461.

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AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.

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6

DRYŁO, Robert. "CONSTRUCTING PAIRING-FRIENDLY GENUS 2 CURVES." National Security Studies 6, no. 2 (December 5, 2014): 95–124. http://dx.doi.org/10.37055/sbn/135218.

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W kryptografii opartej na iloczynach dwuliniowych stosuje się specjalne krzywe, dla których iloczyny dwuliniowe Weila i Tate można efektywnie obliczyć. Takie krzywe, zwykle nazywane pairing-friendly, mają mały stopień zanurzeniowy i wymagają specjalnej konstrukcji. W praktyce stosuje się głównie krzywe eliptyczne i hipereliptyczne genusu 2. Konstrukcje takich krzywych opierają się na metodzie mnożeń zespolonych (CM metodzie) i stąd ograniczają się do krzywych, których pierścień endomorfizmów jakobianu jest generowany przez odpowiednio małe liczby. Aby skonstruować krzywą najpierw wyznacza się parametry jej jakobianu, które zwykle są dane przez liczby Weila dla krzywych genusu 2, a następnie stosuje się CM metodę, aby znaleźć równanie krzywej. Freeman, Scott i Teske zebrali i opisali w ujednolicony sposób metody konstruowania krzywych eliptycznych z danym stopniem zanurzeniowym. Istnieje kilka różnych podejść do konstruowania krzywych genusu 2, z których pierwsze podali Freeman, Stevenhagen i Streng, Kawazoe-Takahashi i Freeman-Satoh. W tym opracowaniu opisujemy podejście oparte na idei autora, w którym wykorzystujemy opowiednie wielomiany wielu zmiennych, aby jako ich wartości otrzymywać liczby Weila odpowiadające jakobianom krzywych genusu 2 z danym stopniem zanurzeniowym. Takie podejście pozwala konstruować zarówno krzywe genusu 2 o jakobianie absolutnie prostym oraz prostym, ale nie absolutnie prostym. Podajemy bezpośrednie wzory, które wyznaczają rodziny parametryczne krzywych genusu 2 z danym stopniem zanurzeniowym.

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7

Markushevich, Dimitri. "Kowalevski top and genus-2 curves." Journal of Physics A: Mathematical and General 34, no. 11 (March 14, 2001): 2125–35. http://dx.doi.org/10.1088/0305-4470/34/11/306.

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8

de Jong, Robin. "Admissible constants for genus 2 curves." Bulletin of the London Mathematical Society 42, no. 3 (February 17, 2010): 405–11. http://dx.doi.org/10.1112/blms/bdp132.

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9

Goren, Eyal Z., and Kristin E. Lauter. "Genus 2 Curves with Complex Multiplication." International Mathematics Research Notices 2012, no. 5 (April 12, 2011): 1068–142. http://dx.doi.org/10.1093/imrn/rnr052.

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10

Baba, Srinath, and Håkan Granath. "Genus 2 Curves with Quaternionic Multiplication." Journal canadien de mathématiques 60, no. 4 (2008): 734. http://dx.doi.org/10.4153/cjm-2009-033-8.

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11

Borówka, Paweł, and Angela Ortega. "Klein coverings of genus 2 curves." Transactions of the American Mathematical Society 373, no. 3 (December 2, 2019): 1885–907. http://dx.doi.org/10.1090/tran/7971.

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12

Hisil, Huseyin, and Craig Costello. "Jacobian Coordinates on Genus 2 Curves." Journal of Cryptology 30, no. 2 (March 15, 2016): 572–600. http://dx.doi.org/10.1007/s00145-016-9227-7.

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13

Kuhn, Robert M. "Curves of Genus 2 with Split Jacobian." Transactions of the American Mathematical Society 307, no. 1 (May 1988): 41. http://dx.doi.org/10.2307/2000749.

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14

McGuire, Gary, and José Felipe Voloch. "Weights in codes and genus 2 curves." Proceedings of the American Mathematical Society 133, no. 8 (March 15, 2005): 2429–37. http://dx.doi.org/10.1090/s0002-9939-05-08027-5.

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15

Kuhn, Robert M. "Curves of genus $2$ with split Jacobian." Transactions of the American Mathematical Society 307, no. 1 (January 1, 1988): 41. http://dx.doi.org/10.1090/s0002-9947-1988-0936803-3.

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16

Brivio, Sonia, and Filippo F. Favale. "Genus 2 curves and generalized theta divisors." Bulletin des Sciences Mathématiques 155 (September 2019): 112–40. http://dx.doi.org/10.1016/j.bulsci.2019.05.002.

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17

Thériault, Nicolas, Jordi Pujolàs, and Josep Miret. "Trisection for supersingular genus $2$ curves in characteristic $2$." Advances in Mathematics of Communications 8, no. 4 (November 2014): 375–87. http://dx.doi.org/10.3934/amc.2014.8.375.

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18

Chiantini, L., C. Ciliberto, and V. Di Gennaro. "The genus of projective curves." Duke Mathematical Journal 70, no. 2 (May 1993): 229–45. http://dx.doi.org/10.1215/s0012-7094-93-07003-2.

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19

GRZEGORCZYK, I., V. MERCAT, and P. E. NEWSTEAD. "STABLE BUNDLES OF RANK 2 WITH FOUR SECTIONS." International Journal of Mathematics 22, no. 12 (December 2011): 1743–62. http://dx.doi.org/10.1142/s0129167x11007434.

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This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve C. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve C of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems. The results can also be expressed in terms of Koszul cohomology and the methods provide a useful tool for the study of the geometry of the moduli space of curves.

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20

Maisner, Daniel, and Enric Nart. "Zeta Functions of Supersingular Curves of Genus 2." Canadian Journal of Mathematics 59, no. 2 (April 1, 2007): 372–92. http://dx.doi.org/10.4153/cjm-2007-016-6.

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AbstractWe determine which isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to k-isomorphism, leading to the same zeta function.

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21

BOXALL, JOHN, DAVID GRANT, and FRANCK LEPRÉVOST. "5-TORSION POINTS ON CURVES OF GENUS 2." Journal of the London Mathematical Society 64, no. 1 (August 2001): 29–43. http://dx.doi.org/10.1017/s0024610701002113.

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Let C be a smooth proper curve of genus 2 over an algebraically closed field k. Fix a Weierstrass point ∞in C(k) and identify C with its image in its Jacobian J under the Albanese embedding that uses ∞ as base point. For any integer N[ges ]1, we write JN for the group of points in J(k) of order dividing N and J*N for the subset of JN of points of order N. It follows from the Riemann–Roch theorem that C(k)∩J2 consists of the Weierstrass points of C and that C(k)∩J*3 and C(k)∩J* are empty (see [3]). The purpose of this paper is to study curves C with C(k)∩J*5 non-empty.

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22

Miret, Josep M., Jordi Pujolàs, and Anna Rio. "Bisection for genus 2 curves in odd characteristic." Proceedings of the Japan Academy, Series A, Mathematical Sciences 85, no. 4 (April 2009): 55–61. http://dx.doi.org/10.3792/pjaa.85.55.

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23

Doerksen, Kevin. "Genus 2 curves with split Jacobians (abstract only)." ACM Communications in Computer Algebra 42, no. 1-2 (July 25, 2008): 49–50. http://dx.doi.org/10.1145/1394042.1394064.

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24

Cavalieri, Renzo, and Nicola Tarasca. "Classes of Weierstrass points on genus $2$ curves." Transactions of the American Mathematical Society 372, no. 4 (May 20, 2019): 2467–92. http://dx.doi.org/10.1090/tran/7626.

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25

Dokchitser, Tim, and Christopher Doris. "3-torsion and conductor of genus 2 curves." Mathematics of Computation 88, no. 318 (November 14, 2018): 1913–27. http://dx.doi.org/10.1090/mcom/3387.

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26

Tarasca, Nicola. "Double Total Ramifications for Curves of Genus 2." International Mathematics Research Notices 2015, no. 19 (December 2, 2014): 9569–93. http://dx.doi.org/10.1093/imrn/rnu228.

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27

Markushevich, D. "Lagrangian families of jacobians of genus 2 curves." Journal of Mathematical Sciences 82, no. 1 (October 1996): 3268–84. http://dx.doi.org/10.1007/bf02362472.

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28

Tang, Chunming, Maozhi Xu, and Yanfeng Qi. "Faster pairing computation on genus 2 hyperelliptic curves." Information Processing Letters 111, no. 10 (April 2011): 494–99. http://dx.doi.org/10.1016/j.ipl.2011.02.011.

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29

Lange, Tanja. "Formulae for Arithmetic on Genus 2 Hyperelliptic Curves." Applicable Algebra in Engineering, Communication and Computing 15, no. 5 (November 12, 2004): 295–328. http://dx.doi.org/10.1007/s00200-004-0154-8.

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30

Riquelme, Edgardo. "Trisection for genus 2 curves in odd characteristic." Applicable Algebra in Engineering, Communication and Computing 27, no. 5 (January 30, 2016): 373–97. http://dx.doi.org/10.1007/s00200-015-0282-3.

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31

Flynn, E. Victor, and Joseph L. Wetherell. "Finding rational points on bielliptic genus 2 curves." manuscripta mathematica 100, no. 4 (December 1, 1999): 519–33. http://dx.doi.org/10.1007/s002290050215.

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32

Bremner, Andrew. "Some Interesting Curves of genus 2 to 7." Journal of Number Theory 67, no. 2 (December 1997): 277–90. http://dx.doi.org/10.1006/jnth.1997.2189.

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33

Booker, Andrew R., Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki. "A database of genus-2 curves over the rational numbers." LMS Journal of Computation and Mathematics 19, A (2016): 235–54. http://dx.doi.org/10.1112/s146115701600019x.

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We describe the construction of a database of genus-$2$curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated$L$-function. This data has been incorporated into the$L$-Functions and Modular Forms Database (LMFDB).

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34

Pujolàs, Jordi, Edgardo Riquelme, and Nicolas Thériault. "Trisection for non-supersingular genus 2 curves in characteristic 2." International Journal of Computer Mathematics 93, no. 8 (July 6, 2015): 1254–64. http://dx.doi.org/10.1080/00207160.2015.1059935.

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35

Fan, Jing, Xuejun Fan, Ningning Song, and Long Wang. "Hyperelliptic Covers of Different Degree for Elliptic Curves." Mathematical Problems in Engineering 2022 (July 4, 2022): 1–11. http://dx.doi.org/10.1155/2022/9833393.

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In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyperelliptic curves to those on elliptic curves. We can also use the cover map as a kind of cover attacks. Given an elliptic curve over an extension field of degree n , one might try to use the cover attack to reduce the discrete logarithm problem (DLP) in the group of rational points of the elliptic curve to DLPs in the Jacobian of a curve of genus g ≥ n over the base field. An algorithm has been proposed for finding genus 3 hyperelliptic covers as a cover attack for elliptic curves with cofactor 2. Our algorithms are about the cover map from hyperelliptic curves of genus 2 to elliptic curves of prime order. As an application, an example of an elliptic curve whose order is a 256-bit prime vulnerable to our algorithms is given.

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36

Bröker, Reinier, and Kristin Lauter. "Modular Polynomials for Genus 2." LMS Journal of Computation and Mathematics 12 (2009): 326–39. http://dx.doi.org/10.1112/s1461157000001546.

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Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.

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37

GUÀRDIA, J. "EXPLICIT GEOMETRY ON A FAMILY OF CURVES OF GENUS 3." Journal of the London Mathematical Society 64, no. 2 (October 2001): 299–310. http://dx.doi.org/10.1112/s0024610701002538.

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An explicit geometrical study of the curves[formula here]is presented. These are non-singular curves of genus 3, defined over ℚ(a). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2-torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curve [Cscr ]1±√2, which is isomorphic to the Fermat curve of degree 4, the computations yield a finer splitting of its jacobian than the classical one.

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38

Miret, Josep, Jordi Pujolàs, and Anna Rio. "Explicit 2-power torsion of genus 2 curves over finite fields." Advances in Mathematics of Communications 4, no. 2 (May 2010): 155–68. http://dx.doi.org/10.3934/amc.2010.4.155.

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39

Cardona, G., J. González, J. C. Lario, and A. Rio. "On curves of genus 2 with Jacobian of GL 2 -type." manuscripta mathematica 98, no. 1 (January 1, 1999): 37–54. http://dx.doi.org/10.1007/s002290050123.

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40

Geer, Gerard Van Der, and Marcel Van Der Vlugt. "Supersingular Curves of Genus 2 over Finite Fields of Characteristic 2." Mathematische Nachrichten 159, no. 1 (1992): 73–81. http://dx.doi.org/10.1002/mana.19921590106.

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41

Pál, Ambrus. "Solvable Points on Projective Algebraic Curves." Canadian Journal of Mathematics 56, no. 3 (June 1, 2004): 612–37. http://dx.doi.org/10.4153/cjm-2004-028-0.

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AbstractWe examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus g, when g is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic p has a divisor of degree prime to 6p defined over a solvable extension.

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42

Lalín, Matilde, and Gang Wu. "Regulator proofs for Boyd’s identities on genus 2 curves." International Journal of Number Theory 15, no. 05 (May 28, 2019): 945–67. http://dx.doi.org/10.1142/s1793042119500519.

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We use the elliptic regulator to recover some identities between Mahler measures involving certain families of genus 2 curves that were conjectured by Boyd and proven by Bertin and Zudilin by differentiating the Mahler measures and using hypergeometric identities. Since our proofs involve the regulator, they yield light into the expected relation of each Mahler measure to special values of [Formula: see text]-functions of certain elliptic curves.

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43

Rains, Eric, and Steven Sam. "Invariant theory of ∧3(9) and genus-2 curves." Algebra & Number Theory 12, no. 4 (July 11, 2018): 935–57. http://dx.doi.org/10.2140/ant.2018.12.935.

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44

Miret, Josep M., Jordi Pujolàs, and Nicolas Thériault. "Bisection for genus 2 curves with a real model." Bulletin of the Belgian Mathematical Society - Simon Stevin 22, no. 4 (November 2015): 589–602. http://dx.doi.org/10.36045/bbms/1447856061.

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45

Flynn, E. Victor, Damiano Testa, and Ronald van Luijk. "Two-coverings of Jacobians of curves of genus 2." Proceedings of the London Mathematical Society 104, no. 2 (July 25, 2011): 387–429. http://dx.doi.org/10.1112/plms/pdr012.

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46

Weng, Annegret. "Constructing hyperelliptic curves of genus 2 suitable for cryptography." Mathematics of Computation 72, no. 241 (May 3, 2002): 435–59. http://dx.doi.org/10.1090/s0025-5718-02-01422-9.

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47

Lauter, Kristin, Michael Naehrig, and Tonghai Yang. "Hilbert theta series and invariants of genus 2 curves." Journal of Number Theory 161 (April 2016): 146–74. http://dx.doi.org/10.1016/j.jnt.2015.02.020.

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48

Neumann, V. G. Lopez, and Constantin Manoil. "Rational classes and divisors on curves of genus 2." manuscripta mathematica 120, no. 4 (July 1, 2006): 403–13. http://dx.doi.org/10.1007/s00229-006-0021-y.

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49

van Wamelen, Paul. "Poonen's question concerning isogenies between Smart's genus 2 curves." Mathematics of Computation 69, no. 232 (August 18, 1999): 1685–98. http://dx.doi.org/10.1090/s0025-5718-99-01179-5.

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50

SCHOEN, CHAD. "A FAMILY OF SURFACES CONSTRUCTED FROM GENUS 2 CURVES." International Journal of Mathematics 18, no. 05 (May 2007): 585–612. http://dx.doi.org/10.1142/s0129167x07004175.

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We consider the deformations of the two-dimensional complex analytic variety constructed from a genus 2 Riemann surface by attaching its self-product to its Jacobian in an elementary way. The deformations are shown to be unobstructed, the variety smooths to give complex projective manifolds whose invariants are computed and whose images under Albanese maps (re)verify an instance of the Hodge conjecture for certain abelian fourfolds.

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