Abstract
We saw in Chap. 11 that Abel had a very ambitious program aiming to study all possible relations between abelian integrals. Even if he did not always completely prove them, he discovered many theorems concerning those relations. For instance, in [115] Kleiman lists and sketches modern proofs of the theorems he encountered in Abel’s paper [1]. One of the most famous such theorems, which is explained in nearly every textbook on algebraic curves and Riemann surfaces, is the following one:
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Notes
- 1.
One may read Catanese’s paper [35] for learning about the relations of such algebraization theorems with Abel’s works.
References
N.H. Abel, Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes, Presented at the French Académie des Sciences in Paris on the 30 October 1826. Republished in Œuvres complètes de Niels Henrik Abel, vol. I, ed. by L. Sylow, S. Lie (Grondahl and Son, Christiania, 1881), pp. 145–211
F. Catanese, From Abel’s heritage: transcendental objects in algebraic geometry and their algebraization, in [127], pp. 349–394
P.A. Griffiths, Variations on a theorem of Abel. Invent. Math. 35, 321–390 (1976)
P.A. Griffiths, The legacy of Abel in algebraic geometry, in [127], pp. 179–205
C. Jacobi, Considerationes generales de transcendentibus abelianis. J. Reine Angew. Math. 9, 394–403 (1832)
S.L. Kleiman, What is Abel’s theorem anyway? in [127], pp. 395–440
O.A. Laudal, R. Piene (eds.), The Legacy of Niels Henrik Abel (Springer, New York, 2004)
D. Mumford, Curves and Their Jacobians (The University of Michigan Press, Ann Arbor, 1975). Republished as an Appendix of The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol. 1358 (Springer, Berlin, 1999)
B. Riemann, Theorie der Abelschen Functionen. J. Reine Angew. Math. 54, 115–155 (1857). French translation: Théorie des fonctions abéliennes. Dans Œuvres mathématiques de Riemann, transl. L. Laugel (Gauthier-Villars, Paris, 1898), pp. 89–164. Reprinted by J. Gabay, Sceaux, 1990
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Popescu-Pampu, P. (2016). A Reinterpretation of Abel’s Works. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_17
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