# Isogenies in Theory and Praxis

## Abstract

We want to give an overview on arithmetical aspects of abelian varieties and their torsion structures, isogenies, and resulting Galois representations. This is a wide and deep territory with a huge amount of research activity and exciting results ranging from the highlights of pure mathematics like the proof of Fermat’s last theorem to stunning applications to public-key cryptography. Necessarily we have to be rather superficial, and thus specialists in the different aspects of the topics may be disappointed. But I hope that for many, and in particular for young researchers, the chapter may serve as an appetizer and will raise interest for a fascinating area of mathematics with many open problems (some are very hard and worth a Fields Medal but others are rather accessible).

The first section of the chapter gives basic notions, definitions, and properties of abelian varieties. Disguised as examples one will find their theory over the complex numbers \(\mathbb{C}\) and the special case of elliptic curves. The second section discusses the situation over finite fields, in particular the role of the Frobenius endomorphism, and over number fields where the most interesting results and challenging conjectures occur. Finally we discuss algorithmic aspects of isogenies, mostly of elliptic curves, and relations to cryptography.

## Keywords

Prime Ideal Elliptic Curve Finite Field Elliptic Curf Abelian Variety## References

- [ACF]H. Cohen, G. Frey (eds.),
*Handbook of Elliptic and Hyperelliptic Curve Cryptography*(CRC, Providence, 2005)Google Scholar - [CL]R. Carls, D. Lubicz, A p-adic quasi-quadratic time point counting algorithm. Int. Math. Res. Not.
**4**, 698–735 (2009)MathSciNetGoogle Scholar - [De]M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hamb.
**14**, 197–272 (1941)CrossRefGoogle Scholar - [Di]C. Diem, An index calculus algorithm for plane curves of small degree, in
*Proceedings of ANTS VII*, ed. by F. Heß, S. Pauli, M. Pohst. Lecture Notes in Computer Science, vol. 4076 (Springer, Berlin, 2006), pp. 543–557Google Scholar - [Fa]G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math.
**73**, 349–366 (1983)CrossRefMATHMathSciNetGoogle Scholar - [FLR]J.-Ch. Faugère, D. Lubicz, D. Robert, Computing modular correspondences for abelian varieties. J. Algebra
**343**, 248–277 (2011)CrossRefMATHMathSciNetGoogle Scholar - [FK1]G. Frey, E. Kani, Curves of genus 2 with elliptic differentials and associated Hurwitz spaces. Cont. Math.
**487**, 33–82 (2009)CrossRefMathSciNetGoogle Scholar - [FK2]G. Frey, E. Kani, Correspondences on hyperelliptic curves and applications to the discrete logarithm, in
*Proceedings of SIIS, Warsaw 2011*, ed. by P. Bouvry, M. Klopotek, F. Leprévost, M. Marciniak, A. Mykowiecka, H. Rybiński. Lecture Notes in Computer Science, vol. 7053 (Springer, Berlin, 2012), pp. 1–19Google Scholar - [FK3]G. Frey, E. Kani, Normal Forms of Hyperelliptic Curves of Genus 3, preprintGoogle Scholar
- [Fr1]G. Frey, On ternary equations of Fermat type and relations with elliptic curves, in
*Modular Forms and Fermat’s Last Theorem*, ed. by G. Cornell, J.H. Silverman, G. Stevens (Springer, New York, 1997), pp. 527–548CrossRefGoogle Scholar - [Fr2]G. Frey, Applications of arithmetical geometry to cryptographic constructions, in
*Proceedings of Finite Fields and Application*(2001), pp. 128–161Google Scholar - [Fr3]G. Frey, Relations between arithmetic geometry and public key cryptography. Adv. Math. Commun.
**4**, 281–305 (2010)CrossRefMATHMathSciNetGoogle Scholar - [GSt]St. Galbraith, A. Stolbunov, Improved algorithm for the isogeny problem for ordinary elliptic curves. Appl. Algebra Eng. Commun. Comput.
**24**, 107–131 (2013)Google Scholar - [GS]P. Gaudry, E. Schost, Hyperelliptic point counting record: 254 bit jacobian, June 2008. http://webloria.loria.fr/~gaudry/record127
- [Ha]C. Hall, An open-image theorem for a general class of abelian varieties. Bull. Lond. Math. Soc.
**43**, 703–711 (2011)CrossRefMATHMathSciNetGoogle Scholar - [He]F. Heß, Computing Riemann–Roch spaces in algebraic function fields and related topics. J. Symb. Comput.
**33**(4), 425–445 (2002)CrossRefMATHGoogle Scholar - [JMV]D. Jao, S.D. Miller, R. Venkatesan, Do all elliptic curves of the same order have the same difficulty of discrete log?, in
*Advances of Cryptology-Asiacrypt 2005*. Lecture Notes in Computer Science, vol. 3788 (Springer, Berlin 2005), pp. 21–40Google Scholar - [JS]D. Jao, V. Soukharev, A subexponential algorithm for evaluating large degree isogenies, in
*Algorithmic Number Theory*(Springer Berlin 2010), pp. 219–233Google Scholar - [K]D. Kohel, Endomorphism rings of elliptic curves over finite fields. Ph.D. thesis, Berkeley, 1996Google Scholar
- [Le]R. Lercier, Algorithmique des courbes elliptiques dans les corps finis. Thèse, LIX-CNRS, 1997Google Scholar
- [LR]D. Lubicz, D. Robert, Computing isogenies between abelian varieties. Compos. Math.
**148**, 1483–1515 (2012)CrossRefMATHMathSciNetGoogle Scholar - [M1]D. Mumford,
*Abelian Varieties*(Oxford University Press, Oxford, 1970)MATHGoogle Scholar - [M2]D. Mumford, On the equations defining abelian varieties I–III. Invent. Math.
**1**, 287–354 (1967); Invent. Math.**3**, 75–135 (1967); Invent. Math.**3**, 215–244 (1967)Google Scholar - [Oe]J. Oesterlé, Versions effectives du théorème de Chebotarev sous l’hypothèse de Riemann généralisée. Astérisque
**61**, 165–167 (1979)MATHGoogle Scholar - [R]K. Ribet, On modular representations of \(G(\bar{\mathbb{Q}}\vert \mathbb{Q})\) arising from modular forms. J. Math.
**100**, 431–476 (1990)MATHMathSciNetGoogle Scholar - [Se1]J.P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math.
**15**, 259–331 (1972)CrossRefMATHMathSciNetGoogle Scholar - [Se2]J.P. Serre, Résumé des cours de 1985–1986 (Annuaire du Collège de France, 1986)Google Scholar
- [S]B. Smith, Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves, in
*Advances in Cryptology: EUROCRYPT 2008, Istanbul*. Lecture Notes in Computer Science, vol. 4965 (2008)Google Scholar - [T]J. Tate, Endomorphisms of abelian varieties over finite fields. Invent. Math.
**2**, 134–144 (1966 )CrossRefMATHMathSciNetGoogle Scholar - [V]J. Vélu, Isogénies entre courbes elliptiques. C.R. Acad. Sci. Paris Ser. A
**273**, 238–241 (1971)Google Scholar