Papers · Preprint · v2
A Conditional Closure of the Strong Birch-Swinnerton-Dyer Conjecture
Strong BSD restated as a fixed point, with each assumed input named and scoped.
In plain words
The Strong BSD formula says two numbers attached to an elliptic curve are equal: one from its L function, one from its arithmetic. The paper puts the two sides in a pair and uses a simple averaging map that replaces both by their mean. The identity holds exactly when the original pair is left unchanged by that map. The paper also says what a closure on richer arithmetic data would need in order to give the same statement, without building one.
The main theorem derives the identity from a named recognition hypothesis. In rank zero and one that hypothesis contains the identity itself, and in rank two or more it is the identity rewritten as a leading term formula. So the result is a conditional translation of Strong BSD into the closure framework, not an independent derivation. The other inputs, local hypotheses and comparisons from the literature, are carried with their exact scope but not used for the identity.
The value is in the accounting. Worked examples show what the inputs see that coarser data miss: for the curves 121a1 and 121b1 the local congruences cannot see the Tamagawa numbers 1 and 2. Two of the inputs match open problems, and the paper says so. The scalar algebra and the closure statements are checked in Lean.
What it shows
- Strong BSD holds exactly when the pair of its two sides is fixed by an averaging closure.
- A conditional theorem from a rank split recognition hypothesis, with its strength stated plainly.
- Examples where the inputs carry information that coarser data miss.
- Lean proofs of the scalar algebra and the closure statements.
What it does not claim
It is not a proof of BSD: the main hypothesis is as strong as the conclusion, and no hypothesis is claimed to be necessary. A leading term identity in rank two or more and main conjectures at additive primes 2 and 3 remain open.
Cite
Tsiokos, I. (2026). A Conditional Closure of the Strong Birch-Swinnerton-Dyer Conjecture. Zenodo. https://doi.org/10.5281/zenodo.23130788