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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.

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2 Answers 2

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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.

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  • $\begingroup$ 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)? $\endgroup$
    – Mathyland
    Jan 8, 2022 at 17:26
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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...

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