The Hasse–Weil conjecture for elliptic curves

The L-series of an elliptic curve over a number field has a meromorphic continuation to ℂ. This is the Hasse–Weil conjecture in the form stated in [Wikipedia], and a weak form of [Gross2011], Conjecture 2.7.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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Cite

@misc{cairn-hasse-weil,
  title        = {The Hasse–Weil conjecture for elliptic curves},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/hasse-weil}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

exists_hasMeromorphicContinuation. The $L$ -series of an elliptic curve over a number field has a meromorphic continuation to $C$ . This is the Hasse--Weil conjecture in the form stated in [Wikipedia], and a weak form of [Gross2011], Conjecture 2.7. It is also what the Birch and Swinnerton-Dyer conjecture rests on, as the hypothesis of [Gross2011], Conjecture 2.10.

exists_hasAnalyticContinuation. The Hasse--Weil conjecture ([Tate1966], p. 416; [Gross2011], Conjecture 2.7): the $L$ -series of an elliptic curve over a number field has an analytic continuation to $C$ .

The $L$ -series $L (E, s)$ of an elliptic curve $E$ over a number field converges absolutely on $Re s > 3/2$ ([Tate1966], p. 416; [Gross2011], Lecture 2, §1). The Hasse--Weil conjecture predicts that it has an analytic continuation to the whole complex plane and satisfies a functional equation ([Tate1966], p. 416; [Gross2011], Conjecture 2.7, stated there for the completed $L$ -function). Only the continuation is stated here. Over $Q$ it follows from the modularity theorem.

[Wikipedia] states the conjecture for the Hasse--Weil zeta function and asks only for a meromorphic continuation, which is equivalent to a meromorphic continuation of $L (E, s)$ . That weaker form is the one the Birch and Swinnerton-Dyer conjecture rests on, so it is recorded here as well. [Gross2011], Conjecture 2.10 takes it as a hypothesis, and only near $s = 1$ . The formulation of the Birch and Swinnerton-Dyer conjecture in FormalConjectures/Millennium/BSD.lean instead asserts a global meromorphic continuation, so it implies exists_hasMeromorphicContinuation below.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.HasseWeil (2 statements).

theorem exists_hasMeromorphicContinuation (E : WeierstrassCurve K) [E.IsElliptic] :
    ∃ L, HasMeromorphicContinuation E L

theorem exists_hasAnalyticContinuation (E : WeierstrassCurve K) [E.IsElliptic] :
    ∃ L, HasAnalyticContinuation E L

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[Tate1966] John Tate. "On the conjectures of Birch and Swinnerton-Dyer and a geometric analog." Seminaire Bourbaki, Vol. 9, Exp. No. 306 (1966), 415-440, numdam
[Gross2011] Benedict H. Gross. "Lectures on the conjecture of Birch and Swinnerton-Dyer." Arithmetic of L-functions, IAS/Park City Math. Ser. 18, AMS (2011), 169-209, math.harvard.edu

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.