A proof of quintic reciprocity using the arithmetic of y² = x⁵ + 1/4

David Grant

Acta Arithmetica (1996)

  • Volume: 75, Issue: 4, page 321-337
  • ISSN: 0065-1036

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David Grant. "A proof of quintic reciprocity using the arithmetic of y² = x⁵ + 1/4." Acta Arithmetica 75.4 (1996): 321-337. <http://eudml.org/doc/206880>.

@article{DavidGrant1996,
author = {David Grant},
journal = {Acta Arithmetica},
keywords = {curves of genus 2; quintic reciprocity law; complex multiplication; Jacobian; explicit computations; formal group},
language = {eng},
number = {4},
pages = {321-337},
title = {A proof of quintic reciprocity using the arithmetic of y² = x⁵ + 1/4},
url = {http://eudml.org/doc/206880},
volume = {75},
year = {1996},
}

TY - JOUR
AU - David Grant
TI - A proof of quintic reciprocity using the arithmetic of y² = x⁵ + 1/4
JO - Acta Arithmetica
PY - 1996
VL - 75
IS - 4
SP - 321
EP - 337
LA - eng
KW - curves of genus 2; quintic reciprocity law; complex multiplication; Jacobian; explicit computations; formal group
UR - http://eudml.org/doc/206880
ER -

References

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  3. [BoBa] J. Boxall et E. Bavencoffe, Quelques propriétés arithmétiques des points de 3-division de la jacobienne de y² = x⁵-1, Séminaire de Théorie des Nombres, Bordeaux 4 (1992), 113-128. Zbl0766.14019
  4. [C] J. W. S. Cassels, On Kummer sums, Proc. London Math. Soc. (3) 21 (1970), 19-27. Zbl0197.32004
  5. [F] R. Fueter, Reziprozitätsgesetze in quadratisch-imaginären Körpern, Nachr. Ges. Wiss. Göttingen 1927, 1-11, 427-445. Zbl54.0192.02
  6. [Gra1] D. Grant, A generalization of a formula of Eisenstein, Proc. London Math. Soc. (3) 62 (1991), 121-132. Zbl0738.14019
  7. [Gra2] D. Grant, Units from 3- and 4-torsion on Jacobians of curves of genus 2, Compositio Math. 95 (1994), 311-320. Zbl0828.11033
  8. [Gra3] D. Grant, Units from 5-torsion on the Jacobian of y² = x⁵ + 1/4 and the conjectures of Stark and Rubin, in preparation. 
  9. [Gra4] D. Grant, Formal groups in genus two, J. Reine Angew. Math. 411 (1990), 96-121. 
  10. [Gra5] D. Grant, Coates-Wiles towers in dimension two, Math. Ann. 282 (1988), 645-666. Zbl0726.11039
  11. [Gre] R. Greenberg, On the Jacobian variety of some algebraic curves, Compositio Math. 42 (1981), 345-359. Zbl0475.14026
  12. [H] D. Hilbert, Théorie des corps de nombres algébriques, Jacques Gabay, Sceaux, 1991. Translation of: Die Theorie der algebraischen Zahlkörper , Jahresber. Deutsch. Math.-Verein 4 (1897), 175-546. 
  13. [IrR] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Grad. Texts in Math. 84, Springer, 1982. 
  14. [Iw] K. Iwasawa, A note on Jacobi sums, Sympos. Math. 15 (1975), 447-459. 
  15. [K1] T. Kubota, Reciprocities in Gauss' and Eisenstein's number fields, J. Reine Angew. Math. 208 (1961), 35-50. Zbl0202.33302
  16. [K2] T. Kubota, An application of the power residue theory to some abelian functions, Nagoya Math. J. 27 (1966), 51-54. Zbl0168.29601
  17. [L] S. Lang, Complex Multiplication, Springer, 1983. Zbl0536.14029
  18. [M] J. Milne, Abelian Varieties, in: G. Cornell and J. Silverman (eds.), Arithmetic Geometry, Springer, New York, 1986, 103-150. Zbl0604.14028
  19. [ST] G. Shimura and Y. Taniyama, Complex Multiplication of Abelian Varieties and its Applications to Number Theory, The Mathematical Society of Japan, 1961. Zbl0112.03502
  20. [W1] A. Weil, La cyclotomie jadis et naguère, Enseign. Math. 20 (1974), 247-263. Zbl0352.12006
  21. [W2] A. Weil, Review of 'Mathematische Werke, by Gotthold Eisenstein,'' Bull. Amer. Math. Soc. 82 (1976), 658-663. 
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  23. [W4] A. Weil, Jacobi sums as Grössencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487-495. Zbl0048.27001

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