[Math] Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 – p^3 $ agree

arithmetic-geometrydiophantine equationselliptic-curvesnt.number-theory

I was playing around with sage, when I found that the Mordell-Weil ranks (over $\mathbb{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few exceptions occur at $p=37$, $p=61$, $p=157$, $p=193$, $\ldots$. This pattern struck me as odd, since the two curves are non-isogenous over the ground field, so why would their ranks be correlated?

After some reflection and further experimentation, I found out that if one looks instead at the $2$-Selmer ranks, there is even a stronger pattern: they seem to agree for all primes $p>2$.

I verified this using the following code, written in sage:

for p in primes(100):
    E1 = EllipticCurve(QQ,[0,p^3])
    E2 = EllipticCurve(QQ,[0,-p^3])
    print("p = "+QQ(p).str()+":"),
    rank1 = E1.selmer_rank()
    rank2 = E2.selmer_rank()
    print([rank1,rank2])

which gives

p = 2: [2, 1] p = 3: [1, 1] p = 5: [1, 1] p = 7: [2, 2] p = 11: [2, 2]
p = 13: [1, 1] p = 17: [1, 1] p = 19: [2, 2] p = 23: [2, 2] p = 29: [1, 1]
p = 31: [2, 2] p = 37: [3, 3] p = 41: [1, 1] p = 43: [2, 2] p = 47: [2, 2]
p = 53: [1, 1] p = 59: [2, 2] p = 61: [3, 3] p = 67: [2, 2] p = 71: [2, 2]
p = 73: [1, 1] p = 79: [2, 2] p = 83: [2, 2] p = 89: [1, 1] p = 97: [1, 1]

I have been trying to prove this by following a case distinction according to the residue class of $p$ modulo $12$, and performing a partial $2$-descent for each case, but I keep getting distracted by the thought that
there must be a neater explanation that I'm missing. Hence my question:

Is there?

Edit: It might be useful to note that similar Sage experiments suggest that also (a) the $2$-Selmer ranks of the elliptic curves $y^2=x^3 \pm p$ and $y^2=x^3 \mp p^5$ agree for all $p>2$ and (b) the $2$-Selmer ranks of the elliptic curves $y^2=x^3 \pm p^2$ and $y^2=x^3 \mp p^4$ agree tot all $p>2$.

In fact, here's a conjecture, also born out by computer experiments, which goes even further and subsumes all cases mentioned before:

Conjecture. Let $a$ be an odd jnteger. Then the $2$-Selmer ranks of the elliptic curves $y^2=x^3 + a$ and $y^2=x^3-a^{-1}$ (which of course is isomorphic to $y^2=x^3 – a^5$) are equal.

We get the original statement with $a=p^3$, we get (a) with $a=\pm p$, and (b) with $a=\pm p^2$.

Best Answer

Your curves seem to be quadratic twists of each other. There are many results in the literature about ranks of such curves. E.g. Ranks of quadratic twists of elliptic curves by Donnelly and others, using Stoll's formula for the size of a Selmer group. Chang (in Note on the rank of quadratic twists of Mordell equations) studied Mordell curves of certain type (your curves are Mordell curves).

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