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Algebraic Geometry

Arithmetic and Diophantine Geometry

14Gxx

[1] Matthew H. Baker, Enrique González-Jiménez, Josep González, and Bjorn Poonen. Finiteness results for modular curves of genus at least 2. Amer. J. Math., 127(6):1325–1387, 2005.
[2] Tatiana Bandman, Gert-Martin Greuel, Fritz Grunewald, Boris Kunyavskiĭ, Gerhard Pfister, and Eugene Plotkin. Identities for finite solvable groups and equations in finite simple groups. Compos. Math., 142(3):734–764, 2006.
[3] Arthur Baragar and Ronald van Luijk. K3 surfaces with Picard number three and canonical vector heights. Math. Comp., 76(259):1493–1498 (electronic), 2007.
[4] M. Borovoi, J.-L. Colliot-Thélène, and A. N. Skorobogatov. The elementary obstruction and homogeneous spaces. Duke Math. J., 141(2):321–364, 2008.
[5] Nigel Boston. Reducing the Fontaine-Mazur conjecture to group theory. In Progress in Galois theory, volume 12 of Dev. Math., pages 39–50. Springer, New York, 2005.
[6] Friederike Brezing and Annegret Weng. Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptogr., 37(1):133–141, 2005.
[7] Ezra Brown, Bruce T. Myers, and Jerome A. Solinas. Hyperelliptic curves with compact parameters. Des. Codes Cryptogr., 36(3):245–261, 2005.
[8] Nils Bruin. Visualising Sha[2] in abelian surfaces. Math. Comp., 73(247):1459–1476 (electronic), 2004.
[9] Patrick Corn. The Brauer-Manin obstruction on del Pezzo surfaces of degree 2. Proc. Lond. Math. Soc. (3), 95(3):735–777, 2007.
[10] Patrick Corn. Tate-Shafarevich groups and K3 surfaces. Math. Comp., To appear, 17 pages, 2007.
[11] Jan Denef and Frederik Vercauteren. An extension of Kedlaya's algorithm to Artin-Schreier curves in characteristic 2. In Algorithmic Number Theory (Sydney, 2002), volume 2369 of Lecture Notes in Comput. Sci., pages 308–323. Springer, Berlin, 2002.
[12] Claus Diem. The GHS attack in odd characteristic. J. Ramanujan Math. Soc., 18(1):1–32, 2003.
[13] Xander Faber and Benjamin Hutz. On the number of rational iterated pre-images of the origin under quadratic dynamical systems. arXiv:0810.1715, 18 pages, 2008.
[14] Xander Faber, Benjamin Hutz, Patrick Ingram, Rafe Jones, Michelle Manes, Thomas J. Tucker, and Michael E. Zieve. Uniform bounds on pre-images under quadratic dynamical systems. Math. Res. Lett., 16(1):87–101, 2009.
[15] Tom Fisher. A new approach to minimising binary quartics and ternary cubics. Math. Res. Lett., 14(4):597–613, 2007.
[16] Tom Fisher. Finding rational points on elliptic curves using 6-descent and 12-descent. J. Algebra, 320(2):853–884, 2008.
[17] E. V. Flynn. The Hasse principle and the Brauer-Manin obstruction for curves. Manuscripta Math., 115(4):437–466, 2004.
[18] David Freeman and Kristin Lauter. Computing endomorphism rings of Jacobians of genus 2 curves over finite fields. In Algebraic geometry and its applications, volume 5 of Ser. Number Theory Appl., pages 29–66. World Sci. Publ., Hackensack, NJ, 2008.
[19] Steven D. Galbraith. Supersingular curves in cryptography. In Advances in Cryptology—Asiacrypt 2001 (Gold Coast), volume 2248 of Lecture Notes in Comput. Sci., pages 495–513. Springer, Berlin, 2001.
[20] Steven D. Galbraith. Weil descent of Jacobians. Discrete Appl. Math., 128(1):165–180, 2003.
[21] Steven D. Galbraith, Florian Hess, and Nigel P. Smart. Extending the GHS Weil descent attack. In Advances in Cryptology—Eurocrypt 2002 (Amsterdam), volume 2332 of Lecture Notes in Comput. Sci., pages 29–44. Springer, Berlin, 2002.
[22] Steven D. Galbraith and Xibin Lin. Computing pairings using x-coordinates only. Des. Codes Cryptogr., 50(3):305–324, 2009.
[23] P. Gaudry, F. Hess, and N. P. Smart. Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptology, 15(1):19–46, 2002.
[24] Pierrick Gaudry. An algorithm for solving the discrete log problem on hyperelliptic curves. In Advances in Cryptology—Eurocrypt 2000 (Bruges), volume 1807 of Lecture Notes in Comput. Sci., pages 19–34. Springer, Berlin, 2000.
[25] Ralf Gerkmann. Relative rigid cohomology and deformation of hypersurfaces. Int. Math. Res. Pap. IMRP, (1):Art. ID rpm003, 67, 2007.
[26] Josep González and Victor Rotger. Non-elliptic Shimura curves of genus one. J. Math. Soc. Japan, 58(4):927–948, 2006.
[27] Cem Güneri, Henning Stichtenoth, and Ihsan Taşkın. Further improvements on the designed minimum distance of algebraic geometry codes. J. Pure Appl. Algebra, 213(1):87–97, 2009.
[28] Johan P. Hansen. Toric varieties, Hirzebruch surfaces and error-correcting codes. Appl. Algebra Engrg. Comm. Comput., 13(4):289–300, 2002.
[29] David Harari and Tamás Szamuely. Galois sections for abelianized fundamental groups. Math. Ann., 344(4):779–800, 2009.
[30] David Harvey. Kedlaya's algorithm in larger characteristic. Int. Math. Res. Not. IMRN, (22):Art. ID rnm095, 29, 2007.
[31] F. Hess. Weil descent attacks. In Advances in Elliptic Curve Cryptography, volume 317 of London Math. Soc. Lecture Note Ser., pages 151–180. Cambridge Univ. Press, Cambridge, 2005.
[32] Florian Hess. A note on the Tate pairing of curves over finite fields. Arch. Math. (Basel), 82(1):28–32, 2004.
[33] Christopher Holden. Mod 4 Galois representations and elliptic curves. Proc. Amer. Math. Soc., 136(1):31–39 (electronic), 2008.
[34] E. W. Howe and K. E. Lauter. Improved upper bounds for the number of points on curves over finite fields. Ann. Inst. Fourier (Grenoble), 53(6):1677–1737, 2003.
[35] Everett W. Howe. Supersingular genus-2 curves over fields of characteristic 3. In Computational arithmetic geometry, volume 463 of Contemp. Math., pages 49–69. Amer. Math. Soc., Providence, RI, 2008.
[36] Everett W. Howe, Kristin E. Lauter, and Jaap Top. Pointless curves of genus three and four. In Arithmetic, Geometry and Coding Theory (AGCT 2003), volume 11 of Sémin. Congr., pages 125–141. Soc. Math. France, Paris, 2005.
[37] Nathan Owen Ilten and Hendrik Süß. AG codes from polyhedral divisors. arXiv:0811.2696, 30 pages, 2008.
[38] Farzali A. Izadi and V. Kumar Murty. Counting points on an abelian variety over a finite field. In Progress in Cryptology—Indocrypt 2003, volume 2904 of Lecture Notes in Comput. Sci., pages 323–333. Springer, Berlin, 2003.
[39] Rafe Jones and Jeremy Rouse. Iterated endomorphisms of Abelian algebraic groups. arXiv:0707.2384, 34 pages, 2007.
[40] Samuel Kadziela. Rigid analytic uniformization of curves and the study of isogenies. Acta Appl. Math., 99(2):185–204, 2007.
[41] Kiran S. Kedlaya. Computing zeta functions via p-adic cohomology. In Algorithmic Number Theory, volume 3076 of Lecture Notes in Comput. Sci., pages 1–17. Springer, Berlin, 2004.
[42] Kenji Koike and Annegret Weng. Construction of CM Picard curves. Math. Comp., 74(249):499–518 (electronic), 2005.
[43] Aristides Kontogeorgis and Victor Rotger. On abelian automorphism groups of Mumford curves and applications to Shimura curves. arXiv:math.AG/0604099, 16 pages, 2006.
[44] Andrew Kresch and Yuri Tschinkel. Integral points on punctured abelian surfaces. In Algorithmic Number Theory (Sydney, 2002), volume 2369 of Lecture Notes in Comput. Sci., pages 198–204. Springer, Berlin, 2002.
[45] Andrew Kresch and Yuri Tschinkel. On the arithmetic of del Pezzo surfaces of degree 2. Proc. London Math. Soc. (3), 89(3):545–569, 2004.
[46] Andrew Kresch and Yuri Tschinkel. Effectivity of Brauer-Manin obstructions. Adv. Math., 218(1):1–27, 2008.
[47] L. Kulesz, G. Matera, and É. Schost. Uniform bounds on the number of rational points of a family of curves of genus 2. J. Number Theory, 108(2):241–267, 2004.
[48] Gilles Lachaud and Christophe Ritzenthaler. On a conjecture of Serre on abelian threefolds. In Algebraic Geometry and its applications, Proceedings od the First SAGA conference, Papeete, France 2007, pages 1–28, 2008.
[49] Alan G. B. Lauder. Counting solutions to equations in many variables over finite fields. Found. Comput. Math., 4(3):221–267, 2004.
[50] Alan G. B. Lauder. A recursive method for computing zeta functions of varieties. LMS J. Comput. Math., 9:222–269 (electronic), 2006.
[51] F. Leprévost, M. Pohst, and A. Schöpp. Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5, 7 or 10. Abh. Math. Sem. Univ. Hamburg, 74:193–203, 2004.
[52] John Little and Hal Schenck. Toric surface codes and Minkowski sums. SIAM J. Discrete Math., 20(4):999–1014 (electronic), 2006.
[53] Adam Logan. The Brauer-Manin obstruction on del Pezzo surfaces of degree 2 branched along a plane section of a Kummer surface. Math. Proc. Cambridge Philos. Soc., 144(3):603–622, 2008.
[54] Michelle Manes. Q-rational cycles for degree-2 rational maps having an automorphism. Proc. Lond. Math. Soc. (3), 96(3):669–696, 2008.
[55] David Savitt. The maximum number of points on a curve of genus 4 over F8 is 25. Canad. J. Math., 55(2):331–352, 2003.
[56] Éric Schost. Computing parametric geometric resolutions. Appl. Algebra Engrg. Comm. Comput., 13(5):349–393, 2003.
[57] R. Shaw. The polynomial degrees of Grassmann and Segre varieties over GF(2). Discrete Math., 308(5-6):872–879, 2008.
[58] Edlyn Teske. An elliptic curve trapdoor system (extended abstract). In High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, volume 41 of Fields Inst. Commun., pages 341–352. Amer. Math. Soc., Providence, RI, 2004.
[59] Ronald van Luijk. Quartic K3 surfaces without nontrivial automorphisms. Math. Res. Lett., 13(2-3):423–439, 2006.
[60] Ronald van Luijk. Cubic points on cubic curves and the brauer-manin obstruction on k3 surfaces. arXiv:0708.2752v1 [math.NT], 17 pages, 2007.
[61] Anthony Várilly-Alvarado and David Zywina. Arithmetic E8 lattices with maximal Galois action. arXiv:0803.3063, 2008.
[62] Eric R. Verheul. Evidence that XTR is more secure than supersingular elliptic curve cryptosystems. J. Cryptology, 17(4):277–296, 2004.
[63] John Voight. Shimura curves of genus at most two. Math. Comp., 78(266):1155–1172, 2009.
[64] Gabor Wiese. Dihedral Galois representations and Katz modular forms. Doc. Math., 9:123–133 (electronic), 2004.

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