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Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it behaves under finite field extensions.

Let $K$ be a number field and let $p$ be a prime number. Suppose that $E_1$ and $E_2$ are elliptic curves over $K$.

Is the set $$\{L/ K \ | \ [L:K] = p, \ rk(E_1(L)) > rk(E_1(K)) \ \textrm{and} \ rk(E_2(L)) > rk(E_2(K)) \}$$ infinite?

That is, is the set of degree $p$ extensions $L$ of $K$ such that the rank of $E_1$ in $L$ jumps and the rank of $E_2$ jumps in $L$ an infinite set?

If $E_1 = E_2$ (Edit: or $E_1$ and $E_2$ are $K$-isogenous), then the answer is positive. A proof is given in https://arxiv.org/abs/1209.0933

Note that the infinitude of the above set is equivalent to the infinitude of the set $$\{ L/K \ | \ [L:K] =p, \ E_1(L)\setminus E_1(K) \neq \emptyset \ \textrm{and} \ E_2(L) \setminus E_2(K) \neq \emptyset\}.$$

Thus, we may also formulate our question as

Is the set of degree $p$ extensions $L/K$ such that $E_1$ has a "new" $L$-point and $E_2$ has a "new" $L$-point infinite?

One may ask of course similar questions about abelian varieties. But I do not even know whether the rank of an abelian variety over $\mathbb{Q}$ jumps in infinitely many quadratic extensions of $\mathbb{Q}$.

Improve this question
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    $\begingroup$ The answer is also positive if the two elliptic curves are isogenous over $K$. $\endgroup$
    – Jason Starr
    Commented Jan 15, 2022 at 0:31
  • $\begingroup$ Yes, good point. The ranks are then equal of course. $\endgroup$
    – Ariyan Javanpeykar
    Commented Jan 15, 2022 at 0:54
  • $\begingroup$ In the linked paper, for $p\geq 5$, the author (Dave Mendes Da Costa) proves the required statement for an elliptic curve $E$ by showing that there is a higher genus curve $C$, a finite morphism $C\to E$ of degree $<p$ and a morphism $C\to \mathbb{P}^1_K$ of degree $p$. The answer to the question would be positive if we could find a higher genus curve $C$ such that there is a finite morphism $C\to E_1$ of degree $<p$, a finite morphism $C\to E_2$ of degree $<p$, and a finite morphism $C\to \mathbb{P}^1_K$ of degree $p$. $\endgroup$
    – Ariyan Javanpeykar
    Commented Jan 15, 2022 at 20:06
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    $\begingroup$ When $p=2$, $K = \mathbb Q$, and $j(E_1)$ and $j(E_2)$ are not simultaneously $0$ or $1728$, then a positive answer follows form Theorem 4 of this paper. $\endgroup$
    – Ariel Weiss
    Commented Feb 7, 2022 at 14:32

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