The rank of elliptic curves and related quadratic twists

Question

Let $E/\mathbb{Q}$ be an elliptic curve, and let $k_1, k_2$ be square-free integers. Can anything be said about the related elliptic curves

$$\displaystyle E/\mathbb{Q}, E^{(k_1)}/\mathbb{Q}, E^{(k_2)}/\mathbb{Q}, E^{(k_1 k_2)}/\mathbb{Q}?$$

Here $E^{(d)}$ is the quadratic twist of $E$ by $d$. If $E$ has the short Weierstrass equation

$$\displaystyle E: y^2 = x^3 + Ax + B, A, B \in \mathbb{Z}$$

then $E^{(d)}$ has the short Weierstrass equation

$$\displaystyle E^{(d)}: dy^2 = x^3 + Ax + B.$$

In particular, let $r(E)$ denote the algebraic rank of $E$ and $r_{\text{an}}(E)$ denote the analytic rank. Are there any relations between the integers

$$\displaystyle r(E), r(E^{(k_1)}), r(E^{(k_2)}), r(E^{(k_1 k_2)})$$

or

$$\displaystyle r_{\text{an}}(E), r_{\text{an}}(E^{(k_1)}), r_{\text{an}}(E^{(k_2)}), r_{\text{an}}(E^{(k_1 k_2)})?$$

Edit:

Observe that if we write

$\displaystyle L(E,s) = \prod_p \left(1 - \frac{\alpha_p}{p^s} \right) \left(1 - \frac{\beta_p}{p^s} \right)$

for the Hasse-Weil $L$-function of $E$, then we have

$\displaystyle L(E,s) L(E^{(k_1 k_2)},s) = \prod_p \left(1 - \frac{\alpha_p}{p^s} \right) \left(1 - \frac{\chi_{k_1 k_2}(p) \alpha_p}{p^s} \right) \left(1 - \frac{\beta_p}{p^s} \right) \left(1 - \frac{\chi_{k_1 k_2}(p) \beta_p}{p^s} \right)$

and

$\displaystyle L(E^{(k_1)}, s) L(E^{(k_2)}, s) = \prod_p \left(1 - \frac{\chi_{k_1}(p) \alpha_p}{p^s} \right)\left(1 - \frac{\chi_{k_2}(p) \alpha_p}{p^s} \right)\left(1 - \frac{\chi_{k_1}(p) \beta_p}{p^s} \right)\left(1 - \frac{\chi_{k_2}(p) \beta_p}{p^s} \right).$

In particular, their Euler factor at $p$ coincides whenever $\chi_{k_1 k_2}(p) = -1$, and whenever $\chi_{k_1 k_2}(p) = \chi_{k_1}(p) = \chi_{k_2}(p) = 1$. Therefore, the Euler factors match for $3/4$'s of primes $p$.

Answer 1

You are looking at the biquadratic extension $K:=\mathbb Q(\sqrt{k_1},\sqrt{k_2})$ of $\mathbb Q$ and want to relate properties of $E(F)$ where $F$ is $\mathbb Q$ or one of the quadratic subfields $F_1,F_2,F_3$ of $K$. If you're also willing to throw $K$ into the mix, then you can use the (unique) idempotent relation on $\text{Gal}(K/\mathbb Q)$ to get a relation on the ranks of $E(K)$, $E(F_1)$, $E(F_2)$, $E(F_3)$, and $E(\mathbb Q)$. This is part of a much larger theory. See for example Kani, Ernst; Rosen, Michael, Idempotent relations among arithmetic invariants attached to number fields and algebraic varieties, J. Number Theory 46, No. 2, 230-254 (1994). ZBL0853.14011.

Answer 2

Let $E, E^{(k)}$ be defined over a number field $K$ and let $k \in K^\times \setminus K^2$. Then it is well-known that $$\operatorname{rk}(E/K(\sqrt d)) = \operatorname{rk}(E/K) + \operatorname{rk}(E^{(k)}/K).$$ Your question is related to the parity conjecture, which asserts that $$w(E/K) = (-1)^{\operatorname{rk}(E/K)},$$ where $w(E/K)$ is the root number of $E$ over $K$. The root number of an elliptic curve can be computed relatively easily. The following two results are taken from this survey paper and might be of interest to you.

Let $E$ be an elliptic curve over $\mathbb Q$. Then the function $k \mapsto w(E^{(k)}/\mathbb Q)$ defined on squarefree integers is periodic, with period depending on the conductor of $E$. $50\%$ of quadratic twists of $E$ have root number $1$ and $50\%$ of quadratic twists of $E$ have root number $-1$.

Answer 3

Here is some experimental data.

For positive integer $k$ let $E_k: y^2=x^3+k x $ and $k_1=2,k_2=3$.

According to computations with sage, for $0 < k < 2000$:

1. At least one of $\displaystyle r_{\text{an}}(E_k), r_{\text{an}}(E_k^{(2)}), r_{\text{an}}(E_k^{(3)}), r_{\text{an}}(E_k^{(6)})$ is positive.

2. At least one of $\displaystyle r_{\text{an}}(E_k), r_{\text{an}}(E_k^{(2)}), r_{\text{an}}(E_k^{(3)}), r_{\text{an}}(E_k^{(6)})$ is odd.

3. For $0 < k < 10^5$ at least one of the root numbers of $\displaystyle (E_k), (E_k^{(2)}), (E_k^{(3)}), (E_k^{(6)})$ is $-1$.

Here is sage code

#Author Georgi Guninski, Mon Aug 12 04:42:25 PM UTC 2024
lim=100 #upper bound for k
for k in range(2,lim):
    Ei=[EllipticCurve([k*m^2,0]) for m in [1,2,3,6]]
    ra=[E.analytic_rank() for E in Ei]
    rap=[i%2 for i in ra]
    print(k,ra,rap)
    assert any(ra) and any(rap)
print("end")