On Silverman's conjecture for a family of elliptic curves

Document Type: Research Paper

Authors

1 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran.

2 Department of Mathematics, Azarbaijan Shahid Madani University, P. O. Box 53751-71379, Tabriz , Iran.

Abstract

Let $E$ be an elliptic curve over $\Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(\Bbb{Q})$ be the group of $\Bbb{Q}$-rational points of $E^{(D)}$.
It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $E^{(p)}(\Bbb{Q})$ has positive rank, and there are infinitely many primes $q$ for which $E^{(q)}(\Bbb{Q})$ has rank $0$. In this paper, assuming the parity conjecture, we show that for infinitely many primes $p$, the elliptic curve $E_n^{(p)}: y^2=x^3-np^2x$ has
odd rank and for infinitely many primes $p$, $E_n^{(p)}(\Bbb{Q})$ has even rank, where $n$ is a positive integer that can be written as biquadrates sums in two different ways, i.e., $n=u^4+v^4=r^4+s^4$, where $u, v, r, s$ are positive integers such that $\gcd(u,v)=\gcd(r,s)=1$. More precisely, we prove that: if $n$ can be written in two different ways as biquartic sums and $p$ is prime, then under the assumption of the parity conjecture $E_n^{(p)}(\Bbb{Q})$ has odd rank (and so a positive rank) as long as $n$ is odd and $p\equiv5, 7\pmod{8}$ or $n$ is even and $p\equiv1\pmod{4}$.
In the end, we also compute the ranks of some specific values of $n$ and $p$ explicitly.

Keywords

Main Subjects


B. J. Birch and N. M. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5 (1966) 295--299.

H. Cohen, Number Theory volume I, Tools and Diophantine Equations, Springer, New York, 2007.

A. Choudhry, The Diophantine equation A4 + B4 = C4 + D4, Indian J. Pure Appl. Math. 22 (1991), no. 1, 9--11.

D. A. Cox, Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication, Pure and Applied Mathematics, John Wiley & Sons, 2011.

J. Cremona, MWRANK Program, available from, http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs/.

L. E. Dickson, History of the Theory of numbers, 2, Reprinted by chelsea, New York, 1971.

L. Euler, Novi Comm. Acad Petrop., v. 17, p. 64. 1772.

L. Euler, Nova Acta Acad, Petrop., v. 13, ad annos 1795-6, 1802, 1778.

L. Euler, Mem. Acad. Sc. St. Petersb., v.11, 1830.

G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Sixth edition. Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles, Oxford University Press, Oxford, 2008.

F. A. Izadi, F. Khoshnam and K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥4, Math. J. Okayama Univ. 56 (2014) 51--63.

G. H. Hardy, Ramanujan: Twelve Lectures on Subjected by His Life and Work, 3rd ed New York, Chelsea, 1999.

L. J. Lander, and T. R. Parkin, Equal sums of biquadrates, Math. Comp. 20 (1966) 450--451.

L. J. Lander, T. R. Parkin, and J. L. Selfridge, A Survey of equal sums of like powers, Math. Comp. 21 (1967) 446--459.

P. Monsky, Mock Heegner Points and Congruent Numbers, Math. Z. 204 (1990), no. 1, 45--67.

K. Ono, Twists Of Elliptic Curves, Compositio Math. 106 (1997), no. 3, 349--360.

K. Ono and T. Ono, Quadratic Form And Elliptic Curves III, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 9, 204--205.

P. Serf, Congruent Numbers and Elliptic Curves, Computational Number Theory (Debrecen, 1989), 227--238, de Gruyter, Berlin, 1991.

SAGE Software, Version 4.3.5, http://sagemath.org.

J. H. Silverman, The Arithmetic of Elliptic curves, Springer-Verlag, New York, 1986.

J. H. Silverman, A Friendly Introduction to Number Theory, Springer-Verlag, New York, 2001.

A. Zajta, Solutions of the Diophantine equation A4 + B4 = C4 + D4, Math. Comp. 41 (1983), no. 164, 635--659.