Skip to main content

Links between solutions of A−B=C and elliptic curves

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1380))

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. BROWNAWELL, W.D., MASSER, D.W.: Vanishing sums in function fields. Math. Proc. Cambr. 1986.

    Google Scholar 

  2. CARAYOL, H.: Sur les représentation 1-adiques associées aux formes modulaires de Hilbert. Ann. Sci. ENS 19, 409–468 (1986).

    MathSciNet  Google Scholar 

  3. CEREDNIK, I.V.: Uniformization of algebraic curves by discrete arithmetic subgroups of PGl2(kw) with compact quotients. Transl. in Math. USSR Sb. 29, 55–78 (1976).

    Article  Google Scholar 

  4. DELIGNE, P.: Preuve des conjectures de Tate et de Shafarevitch (d'après G. Faltings). Sém. Bourbaki 616 (1983).

    Google Scholar 

  5. DELIGNE, P., RAPOPORT, M.: Les schémas de modules de courbes elliptiques. in Modular Functions of One Variable II, Springer Lecture Notes in Math. 349, 143–316 (1972).

    Article  MathSciNet  Google Scholar 

  6. FALTINGS, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73, 349–366 (1983).

    Article  MathSciNet  Google Scholar 

  7. FREY, G.: Some remarks concerning points of finite order on elliptic curves over global fields. Ark. f. Mat. 15, 1–19 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  8. FREY, G.: Rationale Punkte auf Fermatkurven und getwisteten Modulkurven. J. Reine u. Angew. Math. 33, 185–191 (1982).

    MathSciNet  MATH  Google Scholar 

  9. FREY, G.: Links between stable elliptic curves and certain diophantine equations. Ann. Univ. Sarav. Math. Ser. Vol. 1, 1–40 (1986).

    MathSciNet  MATH  Google Scholar 

  10. HELLEGOUARCH, Y.: Points d'ordre 2ph sur les courbes elliptiques. Acta Arith. 26, 253–263 (1975).

    MathSciNet  MATH  Google Scholar 

  11. JORDAN, B., LIVNÉ, R.: Local diophantine properties of Shimura curves. Math. Ann. 270, 235–248 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  12. KATZ, N., MAZUR, B.: Arithmetic moduli of elliptic curves. Princeton Univ. Press (1985).

    Google Scholar 

  13. MAZUR, B.: Modular curves and the Eisenstein ideal. Publ. Math. IHES 47, 33–186 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  14. MAZUR, B.: Rational isogenies of prime degree. Invent. Math. 44, 129–162 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  15. MAZUR, B.: Letter to J.F. Mestre (16. August 1985).

    Google Scholar 

  16. Modular Functions of One Variable IV. Springer Lecture Notes in Math. 476 (1975).

    Google Scholar 

  17. RIBET, K.: Mod p Hecke operators and congruences between modular forms. Invent. Math. 71, 193–205 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  18. RIBET, K.: Congruence relations between modular forms. Proc. Int. Congr. Math., 503–514 (1983).

    Google Scholar 

  19. RIBET, K.: On modular representations of Gal arising from modular forms. Math. Sc. Research Institute Berkeley, CA, Preprint # 06420-87 (1987).

    Google Scholar 

  20. ROQUETTE, P.: Analytic theory of elliptic functions over local fields. Hamb. Math. Einzelschriften, N.F. Heft 1 (1969).

    Google Scholar 

  21. SERRE, J.P.: Sur les représentations modulaires de degré 2 de Gal . Duke Math. J. 54, 179–230 (1987).

    Article  MathSciNet  Google Scholar 

  22. SHIMURA, G.: Introduction to the arithmetic of automorphic functions. Princeton Univ. Press (1971).

    Google Scholar 

  23. SILVERMAN, J.H.: The arithmetic of elliptic curves. New York-Berlin-Heidelberg-Tokyo (1986).

    Google Scholar 

  24. STEWART, C.L., TIJDEMAN, R.: On the Oesterlé-Masser conjecture. Monatsh. f. Math. 102, 251–257 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  25. TANIYAMA, Y.: in: Problem session of the Tokyo-Nikko conference on number theory; problem 12, 1955.

    Google Scholar 

  26. TATE, J.: Algorithm for finding the type of a singular fiber in an elliptic pencil. in [16], 33–52.

    Google Scholar 

  27. VOJTA, P.: Diophantine approximation and value distribution theory. Springer Lecture Notes in Math. 1239 (1987).

    Google Scholar 

  28. WEIL, A.: Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168, 149–156 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  29. FALTINGS, G.: Calculus on arithmetical surfaces. Ann. Math. 119, 387–424, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  30. MIYAOKA, Y.: On the Chern numbers of surfaces of general type. Inv. Math. 42, 225–237 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  31. PARSHIN, A.N.: The Bogomolov-Miyaoka-Yau inequality for the arithmetical surfaces and its applications. Preprint 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Hans Peter Schlickewei Eduard Wirsing

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Frey, G. (1989). Links between solutions of A−B=C and elliptic curves. In: Schlickewei, H.P., Wirsing, E. (eds) Number Theory. Lecture Notes in Mathematics, vol 1380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086544

Download citation

  • DOI: https://doi.org/10.1007/BFb0086544

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51397-1

  • Online ISBN: 978-3-540-46205-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics