# Conjectured Diophantine Estimates on Elliptic Curves

• Serge Lang
Chapter
Part of the Progress in Mathematics book series (PM, volume 35)

## Abstract

Let A be an elliptic curve defined over the rational numbers Q. Mordell’s theorem asserts that the group of points A(Q) is finitely generated. Say {P 1,..., P r } is a basis of A(Q) modulo torsion. Explicit upper bounds for the heights of elements in such a basis are not known. The purpose of this note is to conjecture such bounds for a suitable basis. Indeed, RA(Q) is a vector space over R with a positive definite quadratic form given by the Néron-Tate height: if A is defined by the equation
$${y^2} = {x^3} + ax + b,a,b \in {\text{Z}}$$
, and P = (x, y) is a rational point with x = c/d written as a fraction in lowest form, then one defines the x-height
$${h_x} = {\text{log max}}\left( {\left| c \right|,\left| d \right|} \right)$$
.

## Keywords

Zeta Function Elliptic Curve Elliptic Curf Riemann Zeta Function Riemann Hypothesis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [B-C-II-S]
B. BIRCH, S. CHOWLA, M. HALL, A. SCHINZEL, On the difference x3–y2, Norske Vid. Selsk. Forrh. 38 (1965), pp. 65–69.
2. [B-S]
B. BIRCH and N. STEPHENS, The parity of the rank of the Mordell-Weil group, Topology 5 (1966) pp. 295–299.
3. [Ca]
J. W. CASSELS, The rational solutions of the Diophantine equation Y2 = X3 — D, Acta Math. 82 (1950) pp. 243–273. Addenda and corrigenda to the above, Acta Math. 84 (1951), p. 299.
4. [Da]
H. DAVENPORT, On f 3 (t) — g 2 (t), Kon. Norsk Vid. Selsk. For. Bd. 38 Nr. 20 (1965) pp. 86–87.
5. [De]
V. A. DEMJANENKO, Estimate of the remainder term in Tate ‘s formula, Mat. Zam. 3 (1968), pp. 271–278.
6. [Fe]
N. I. FELDMAN, An effective refinement of the exponent in Liouville’s theorem, Izv. Akad. Nauk 35 (1971) pp. 973990, AMS Transl. (1971), pp. 985–1002.Google Scholar
7. [H]
M. HALL, The diophantine equation x3–y2 = k, Computers in Number Theory, Academic Press (1971), pp. 173–198.Google Scholar
8. [L 1]
S. LANG, Elliptic curves: diophantine analysis, Springer Verlag, 1978.Google Scholar
9. [L 2]
S. LANG, On the zeta function of number fields, Invent. Math. 12 (1971), pp. 337–345.
10. [M]
J. MANIN, Cyclotomic fields and modular curves, Russian Mathematical Surveys Vol. 26 No. 6, Published by the London Mathematical Society, Macmillan Journals Ltd, 1971Google Scholar
11. [Mo]
H. MONTGOMERY, Extreme values of the Riemann zeta function, Comment. Math. Helvetici 52 (1977), pp. 511–518.
12. [Po]
V. D. PODSYPANIN, On the equation x 3 = y2 + Az 6,Math. Sbornik 24 (1949), pp. 391–403 (See also Cassels’ corrections in [Ca]).Google Scholar
13. [Sch]
W. SCHMIDT, Thue’s equation over function fields, J. Australian Math. Soc. (A) 25 (1978) pp. 385–422.
14. [Se 1]
E. SELMER, The diophantine equation axa + by 3 + cz 3, Acta Math. (1951), pp. 203–362.Google Scholar
15. [Se 2]
E. SELMER, Ditto, Completion of the tables, Acta. Math. (1954), pp. 191–197.Google Scholar
16. [Sie]
C. L. SIEGEL, Abschätzung von Einheiten, Nachr. Wiss. Göttingen (1969), pp. 71–86.Google Scholar
17. [Sil 1]
J. SILVERMAN, Lower bound for the canonical height on elliptic curves, Duke Math. J. Vol. 48 No. 3 (1981), pp. 633–648.
18. [Sil 2]
J. SILVERMAN, Integer points and the rank of Thue Elliptic curves, Invent. Math. 66 (1962), pp. 395–404.Google Scholar
19. [Sil 3]
J. SILVERMAN, Heights and the Specialization map for families of abelian varieties,to appear.Google Scholar
20. [St]
H. STARK, Effective estimates of solutions of some Diophantine Euations, Acta. Arith. 24 (1973), pp. 251–259.
21. [Ste]
N. STEPHENS, The diophantine equation X 3 +Y 3 = DZ 3 and the conjectures of Birch-Swinnerton Dyer, J. reine angew. Math. 231 (1968), pp. 121–162.
22. [Ta 1]
J. TATE, The arithmetic of elliptic curves, Invent. Math. (1974), pp. 179–206.Google Scholar
23. [Ta 2]
J. TATE, Algorithm for determining the Type of a Singular Fiber in an Elliptic Pencil, Modular Functions of One Variable IV, Springer Lecture Notes 476 (Antwerp Conference) (1972), pp. 33–52.
24. [Ti]
E. C. TITCHMARSH, The theory of the Riemann zeta function, Oxford Clarendon Press, 1951.Google Scholar
25. [Zi]
H. ZIMMER, On the difference of the Weil height and the NéronTate height, Math. Z. 147 (1976), pp. 35–51.