# Rank of the Elliptic Curve $y^2 = x^3 - px$

I found in Silverman's "The Arithmetic of Elliptic Curves," Chapter X, Proposition 6.2. that the rank of $$y^2 = x^3 + Dx$$ is zero when $$D$$ is a prime congruent to 7 mod 16, but I can't find much about $$y^2 = x^3 - Dx$$. Specifically, I'm trying to figure out whether or not the rank of $$y^2 = x^3 - px$$ is $$1$$ for every prime $$p \equiv 7$$ (mod 16). Empirically, SageMath seems to confirm this (I've checked up to $$p=2423$$), but I haven't been able to find a proof of it. If someone could find a proof or give me some direction on this (or a proof that the rank $$\geq$$ 1), it would be greatly appreciated.

I don't know much about the theory of elliptic curves, but I'm doing a related project that uses some results about elliptic curves. So I'm mainly interested in whether or not a proof exists, and I likely wouldn't understand the proof itself.

Here's my SageMath code:

sage: for prime in Primes():
....:     if prime % 16 == 7:
....:         E = EllipticCurve([-prime,0])
....:         rank = E.rank()
....:         print("p = " + str(prime) + ", rank = " + str(rank))
....:         print("")
....:     if prime > 10000:
....:         break
....:
p = 7, rank = 1

p = 23, rank = 1

p = 71, rank = 1

p = 103, rank = 1

p = 151, rank = 1

p = 167, rank = 1

p = 199, rank = 1

p = 263, rank = 1

p = 311, rank = 1

p = 359, rank = 1

p = 439, rank = 1

p = 487, rank = 1

p = 503, rank = 1

p = 599, rank = 1

p = 631, rank = 1

p = 647, rank = 1

p = 727, rank = 1

p = 743, rank = 1

p = 823, rank = 1

p = 839, rank = 1

p = 887, rank = 1

p = 919, rank = 1

p = 967, rank = 1

p = 983, rank = 1

p = 1031, rank = 1

p = 1063, rank = 1

p = 1223, rank = 1

p = 1303, rank = 1

p = 1319, rank = 1

p = 1367, rank = 1

p = 1399, rank = 1

p = 1447, rank = 1

p = 1511, rank = 1

p = 1543, rank = 1

p = 1559, rank = 1

p = 1607, rank = 1

p = 1783, rank = 1

p = 1831, rank = 1

p = 1847, rank = 1

p = 1879, rank = 1

p = 2039, rank = 1

p = 2087, rank = 1

p = 2311, rank = 1

p = 2423, rank = 1


Additionally, if someone has time, I have similar questions about $$y^2 = x^3 + p^3x$$ and $$y^2 = x^3 - p^3x$$ for $$p \equiv 7$$ (mod 16). The former seems to be rank zero and the latter rank $$1$$, but I can't find a proof of either.

There are a few things that might help resolve this problem. One is that using a 2-descent it is possible to find a straightforwad upper bound on the rank of the curve over $$\mathbb{Q}$$. If we let $$E_D : y^2 = x^3 + Dx$$ be our elliptic curve with $$D$$ a 4-th power free integer, Silverman's Proposition X.6.1 shows that the rank of $$E_D(\mathbb{Q})$$ is no more than $$2\nu(2D)-1$$, where $$\nu(2D)$$ is the number of distinct prime divisors of $$2D$$. When $$D=\pm p$$ or $$D=\pm p^3$$ for an odd prime number $$p$$, this upper bound is $$3$$.

However, this bound can be improved by results of Aguirre, Lozano-Robledo, and Peral, in "Elliptic curves of maximal rank," Proceedings of the `Segundas Jornadas de Teoría de Números,' Bibl. Rev. Mat. Iberoamericana (2008), 1-28. One thing they show is that for an elliptic curve $$E : y^2 = x^3 + Ax^2 + Bx, \quad A,\, B \in \mathbb{Z},$$ there is an upper bound, $$\mathrm{rank} (E(\mathbb{Q})) \leq v(A^2-4B) + \nu(B) - 1.$$ For $$E_{\pm p}$$ and $$E_{\pm p^3}$$, this upper bound is now $$2$$.

The next thing to consider is the sign of the functional equation of the $$L$$-function of $$E_D$$. If we let $$r_D$$ be the rank of $$E_D(\mathbb{Q})$$, then Birch and Stephens, "The parity of the rank of the Mordell-Weil group," Topology 5 (1966), 295-299, showed that, assuming the Tate-Shafarevich group of $$E_D/\mathbb{Q}$$ is finite, $$(-1)^{r_D}$$ equals the sign of the function equation of its $$L$$-function. (They assumed further that $$D$$ is not divisible by 4.) They also gave a formula for this sign, and again assuming the finiteness of the Tate-Shafarevich group, they obtained $$(-1)^{r_D} = w_{\infty} w_2 \prod_{p^2\, \lVert\, D} w_p,$$ where $$w_\infty$$ is the sign of $$D$$; $$w_2=-1$$ for $$D \equiv 3, 5, 13, 15 \pmod{16}$$ and $$w_2=1$$ otherwise; and for odd primes, $$w_p=-1$$ for $$p \equiv 3 \pmod{4}$$ and $$w_p=1$$ otherwise. Here $$p^2 \,\lVert\,D$$ means $$p^2$$ divides $$D$$, but $$p^3$$ does not.

In particular, for the curve $$E_{-p}$$ with $$p \equiv 7 \pmod{16}$$, this sign is $$-1$$, so the rank is odd. The upper bound on the rank by Aguirre, Lozano-Robledo, and Peral then implies that the rank must be $$1$$. Similarly when $$p \equiv 7 \pmod{16}$$, the sign of the functional equation for $$E_{-p^3}$$ is also $$-1$$, implying that the rank must be $$1$$. As before, all of these statements about the rank are contingent on the finiteness of the Tate-Shafarevich group. Aguirre, Lozano-Robledo, and Peral give additional illuminating examples of these types of calculations.

• Thank you for your answer; it really helps. So if I’m understanding correctly, we can find from this that $r_{p^3}$ is even and $r_{p^3} \leq 2$, but we would need something else to prove it’s $0$ (assuming that is the case)? Jan 8, 2022 at 17:26

I found it! In the paper "Three constructions of rational points on $$Y^ 2 = X^3 \pm NX$$," P. Monsky constructs a rational point of infinite order on the ellitpic curve $$y^2 = x^3 - Nx$$ for $$N = p$$ and $$N = p^3$$ where $$p$$ is a prime congruent to 7 mod 16. Thus their rank is at least $$1$$. Turns out it's rather non-trivial and uses modular functions.

Now the only thing left is to find out if $$y^2 = x^3 + p^3x$$ is rank zero. Based on how deep the other two results were, I might move this to MathOverflow...