Infinitely many rational elliptic curves with 2-torsion over quadratic fields

About 16 years old · traced to 2010

Let $p$ be a prime satisfying

p\equiv 11,19\pmod{20}.

For an elliptic curve $E/\mathbb Q$ , write $E(\mathbb Q(\sqrt{-p}))_{\mathrm{tors}}$ for its torsion subgroup over $\mathbb Q(\sqrt{-p})$ . Existence conjecture. There exist infinitely many elliptic curves $E/\mathbb Q$ such that

E(\mathbb Q(\sqrt{-p}))_{\mathrm{tors}}\cong \mathbb Z/2\mathbb Z\times\mathbb Z/10\mathbb Z.

This is one of the open questions left by the paper concerning prescribed torsion subgroups of rational elliptic curves over quadratic fields; the paper proves related non-existence results and establishes the asserted infinitude conditionally on finiteness of the $2$ -primary part of the Tate–Shafarevich group.

References

Primary source

Omer Avci, “Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields”, arXiv:2602.07723 (2026).

Additional references

6 papers in this index state this conjecture (2010–2026). The statement above is taken from the most recent of them; the others are arXiv:2201.03452, arXiv:1601.05032, arXiv:1301.6999, arXiv:1201.6629, arXiv:1009.5389.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A 2026 paper proves the expected infinitude only under a standard conjectural assumption, so the unconditional question remains open.

The conjecture, recorded as Conjecture 1.5 in a 2026 paper, asks whether infinitely many rational elliptic curves have the specified torsion over $\mathbb{Q}(\sqrt{-p})$ for primes $p\equiv 11,19\pmod{20}$ .

February 2026 conditional result

The paper studies $X_1(2,10): y^2=x(x^2+x-1)$ and states that, conditional on the parity conjecture, the relevant quadratic twist has rank $1$ for these primes; this yields the conjectured infinitude. It further says that finiteness of the $2$ -primary part of the Tate–Shafarevich group implies the needed parity statement. Unconditionally, it proves non-existence for $p\equiv 3,7\pmod{20}$ , while the target cases remain unresolved.

Current status (as of August 2026): The conjecture has a conditional proof assuming parity, or finiteness of the relevant $2$ -primary Tate–Shafarevich groups, but no unconditional proof or refutation is reported.

Sources

No solutions have been posted yet.