# zbMATH — the first resource for mathematics

On the equation $$Y^ 2=(X+p)(X^ 2+p^ 2)$$. (English) Zbl 0810.11038
The family of elliptic curves over $$\mathbb{Q}$$ given by the title equation is studied. It is shown that for $$p$$ a prime number $$\equiv\pm 3\bmod 8$$, the only rational solution to the equation given here is the one with $$y=0$$. The same is true for $$p=2$$. Standard conjectures predict that the rank of the group of rational points is odd for all other primes $$p$$. A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers $$p$$. Moreover, this bound can only be attained for certain special prime numbers $$p\equiv 1\bmod 16$$. Examples of such rank 3 curves are given. Lastly for certain primes $$p\equiv 9\bmod 16$$ non-trivial elements in the Shafarevich group of the elliptic curve are constructed. In the literature one finds similar investigations of elliptic curves depending on a prime number $$p$$; however, usually these are elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication.

##### MSC:
 11G05 Elliptic curves over global fields 11D25 Cubic and quartic Diophantine equations
Full Text:
##### References:
  B.J. Birch, Heegner points of elliptic curves , Symp. Math. Inst. Alta Math. 15 (1975), 441-445. · Zbl 0317.14015  B.J. Birch and W. Kuyk, Modular functions of one variable IV, Springer-Verlag LNM 476 (1975). · Zbl 0315.14014  A. Bremner, On the equation $$Y^2=X(X^2+p)$$ , in Number theory and applications (R.A. Mollin, ed.), Kluwer, Dordrecht, The Netherlands, 1989, 3-23.  A. Bremner and D. Buell, Three points of great height on elliptic curves , Math. Comp. 61 (1993), 111-115. JSTOR: · Zbl 0785.11035  A. Bremner and J.W.S. Cassels, On the equation $$Y^2=X(X^2+p)$$ , Math. Comp. 42 (1984), 257-264. JSTOR: · Zbl 0531.10014  J. Coates, Elliptic curves and Iwasawa theory , in Modular forms (R.A. Rankin, ed.), Ellis Horwood, Chichester, 1984, 51-73. · Zbl 0561.14012  B. Gross and D. Zagier, Heegner points and derivatives of $$L$$-series , Invent. Math. 84 (1986), 225-320. · Zbl 0608.14019  T. Honda and I. Miyawaki, Zeta-functions of elliptic curves of $$2$$-power conductor , J. Math. Soc. Japan 26 (1974), 362-373. · Zbl 0273.14007  N. Koblitz, Introduction to elliptic curves and modular forms , Springer-Verlag, New York, 1984. · Zbl 0553.10019  J.-F. Mestre, Formules explicites et minorations de conducteurs de variétés algébriques , Compositio Math. 58 (1986), 209-232. · Zbl 0607.14012  A. Ogg, Abelian curves of $$2$$-power conductor , Proc. Cambr. Phil. Soc. 62 (1966), 143-148. · Zbl 0163.15403  J.H. Silverman, The arithmetic of elliptic curves , Springer-Verlag, New York, 1986. · Zbl 0585.14026  ——–, The difference between the Weil height and the canonical height on elliptic curves , Math. Comp. 55 (1990), 723-743. JSTOR: · Zbl 0729.14026  P. Stevenhagen, Divisibility by $$2$$-powers of certain quadratic class numbers , J. Number Theory 43 (1993), 1-19. · Zbl 0767.11054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.