Isogenous components of Jacobian surfaces
Tóm tắt
Let
be a genus 2 curve defined over a field K,
$$\mathrm{char}\,K = p \geqslant 0$$
, and
its Jacobian, where
$$\iota $$
is the principal polarization of
attached to
. Assume that
is (n, n)-geometrically reducible with
$$E_1$$
and
$$E_2$$
its elliptic components. We prove that there are only finitely many curves
(up to isomorphism) defined over K such that
$$E_1$$
and
$$E_2$$
are N-isogenous for
$$n=2$$
and
$$N=2,3, 5, 7$$
with
or
$$n = 2$$
,
$$N = 3,5, 7$$
with
. The same holds if
$$n=3$$
and
$$N=5$$
. Furthermore, we determine the Kummer and Shioda–Inose surfaces for the above
and show how such results in positive characteristic
$$p>2$$
suggest nice applications in cryptography.
Tài liệu tham khảo
Beshaj, L.: Minimal integral Weierstrass equations for genus 2 curves. In: Malmendier, A., Shaska, T. (eds.) Higher Genus Curves in Mathematical Physics and Arithmetic Geometry. Contemporary Mathematics, vol. 703, pp. 63–82. American Mathematical Society, Providence (2018)
Beshaj, L.: Absolute reduction of binary forms. Albanian J. Math. 12(1), 36–77 (2018)
Clingher, A., Doran, C.F.: Modular invariants for lattice polarized \(K3\) surfaces. Mich. Math. J. 55(2), 355–393 (2007)
Costello, C.: Computing supersingular isogenies on Kummer surfaces. In: Peyrin, T., Galbraith, S. (eds.) Advances in Cryptology—ASIACRYPT 2018. Part III. Lecture Notes in Computer Science, vol. 11274, pp. 428–456. Springer, Cham (2018)
Dolgachev, I., Lehavi, D.: On isogenous principally polarized abelian surfaces. In: Alexeev, V., et al. (eds.) Curves and Abelian Varieties. Contemporary Mathematics, vol. 465, pp. 51–69. American Mathematical Society, Providence (2008)
Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)
Frey, G.: On elliptic curves with isomorphic torsion structures and corresponding curves of genus \(2\). In: Coates, J., Yau, S.-T. (eds.) Elliptic Curves, Modular Forms, & Fermat’s Last Theorem. Series in Number Theory, vol. I, pp. 79–98. International Press, Cambridge (1995)
Frey, G., Kani, E.: Curves of genus 2 with elliptic differentials and associated Hurwitz spaces. In: Lachaud, G., Ritzenthaler, C., Tsfasman, M.A. (eds.) Arithmetic, Geometry, Cryptography and Coding Theory. Contemporary Mathematics, vol. 487, pp. 33–81. American Mathematical Society, Providence (2009)
Frey, G., Shaska, T.: Curves, Jacobians, and cryptography. In: Beshaj, L., Shaska, T. (eds.) Algebraic Curves and Their Applications. Contemporary Mathematics, vol. 724, pp. 279–344. American Mathematical Society, Providence (2019)
Hashimoto, K., Murabayashi, N.: Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two. Tohoku Math. J. 47(2), 271–296 (1995)
Igusa, J.: Arithmetic variety of moduli for genus two. Ann. Math. 72, 612–649 (1960)
Kodaira, K.: On compact analytic surfaces. II, III. Ann. Math. 77,78, 563–626, 1–40 (1963)
Kumar, A.: Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields. Res. Math. Sci. 2, # 24 (2015)
Kumar, A., Kuwata, M.: Elliptic K3 surfaces associated with the product of two elliptic curves: Mordell–Weil lattices and their fields of definition. Nagoya Math. J. 228, 124–185 (2017)
Lombardo, D.: Computing the geometric endomorphism ring of a genus-2 Jacobian. Math. Comput. 88(316), 889–929 (2019)
Lubicz, D., Robert, D.: Computing separable isogenies in quasi-optimal time. LMS J. Comput. Math. 18(1), 198–216 (2015)
Magaard, K., Shaska, T., Völklein, H.: Genus 2 curves that admit a degree 5 map to an elliptic curve. Forum Math. 21(3), 547–566 (2009)
Malmendier, A., Shaska, T.: The Satake sextic in F-theory. J. Geom. Phys. 120, 290–305 (2017)
Malmendier, A., Shaska, T.: A universal genus-two curve from Siegel modular forms. SIGMA Symmetry Integrability Geom. Methods Appl. 13, 089 (2017)
Malmendier, A., Shaska, T.: From hyperelliptic to superelliptic curves. Albanian J. Math. 13(1), 107–200 (2019)
Shaska, T.: Curves of genus 2 with \((N, N)\) decomposable Jacobians. J. Symbolic Comput. 31(5), 603–617 (2001)
Shaska, T.: Curves of Genus Two Covering Elliptic Curves. Ph.D. Thesis, University of Florida (2001)
Shaska, T.: Genus 2 curves with \((3,3)\)-split Jacobian and large automorphism group. In: Fieker, C., Kohel, D.R. (eds.) Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 2369, pp. 205–218. Springer, Berlin (2002)
Shaska, T.: Genus 2 fields with degree 3 elliptic subfields. Forum Math. 16(2), 263–280 (2004)
Shaska, T.: Genus two curves covering elliptic curves: a computational approach. In: Shaska, T. (ed.) Computational Aspects of Algebraic Curves. Lecture Notes Series on Computing, vol. 13, pp. 206–231. World Scientific, Hackensack (2005)
Shaska, T.: Genus two curves with many elliptic subcovers. Commun. Algebra 44(10), 4450–4466 (2016)
Shaska, T., Völklein, H.: Elliptic subfields and automorphisms of genus 2 function fields. In: Christensen, C., Sathaye, A., Bajaj, C. (eds.) Algebra, Arithmetic and Geometry with Applications, pp. 703–723. Springer, Berlin (2004)
Shioda, T., Inose, H.: On singular \(K3\) surfaces. In: Baily Jr., W.L., Shioda, T. (eds.) Complex Analysis and Algebraic Geometry, pp. 119–136. Iwanami Shoten, Tokyo (1977)
Zarhin, Yu.G.: Families of absolutely simple hyperelliptic Jacobians. Proc. London Math. Soc. 100(1), 24–54 (2010)