Chowla’s non-vanishing conjecture over Fq(T)

Chowla’s function-field non-vanishing conjecture, interpreted as non-vanishing for every quadratic character over $\mathbb{F}_q(T)$, is false: infinitely many central vanishing examples are known. The remaining question concerns how sparse those exceptions are.

Known results

• Li proved that infinitely many quadratic $L$-functions vanish at the central point.
• Bui and Florea showed that more than $94\%$ of the hyperelliptic family is non-vanishing.
• Ellenberg, Li, and Shusterman obtained a vanishing proportion tending to $0$ as $q\to\infty$.
• Geometric criteria relate central vanishing to maps from $y^2=D$ to a fixed abelian variety.

September 2026 density-one result

Koymans, Pagano, and Shusterman report a density-one theorem: the non-vanishing proportion tends to $1$ in the successive limit $g\to\infty$, then $q\to\infty$, with an explicit finite-$q$ lower bound. This is claimed progress, not non-vanishing for every character.

Current status (as of September 2026): The universal assertion is refuted by infinitely many exceptions, while density-one non-vanishing is claimed in the stated double-limit regime; stronger fixed-$q$ and every-character statements remain open.