Disegni–Zhang conjecture on nonvanishing of local relative characters
For a nonarchimedean local field of characteristic zero and the relevant pair of representations of , the associated local relative character is nonzero: .
References
Primary source
Additional references
- Rankin--Selberg integrals of opposite conductor--one newforms — arXiv — Dongwen Liu, Lei Zhang
Progress summary
A new preprint claims a proof in one restricted setting, giving limited progress but not settling the conjecture.
The Disegni–Zhang conjecture predicts nonvanishing of certain local relative characters. The newly reported work studies a conductor-one case through explicit calculations with opposite newforms.
September 10, 2026 conductor-one case
Dongwen Liu and Lei Zhang’s preprint claims to prove the conjecture in a conductor-one local setting using Rankin–Selberg integrals of opposite conductor-one newforms. This establishes a concrete new family of supporting cases, while leaving the general conjecture open. A separate source continues to use the conjecture as an assumption rather than proving it.
Current status (as of September 2026): A conductor-one local case is claimed, but the general Disegni–Zhang conjecture remains open and the new result is not yet independently verified.
Sources
- arxiv.org
- arxiv.org
- disegni-daniel.perso.math.cnrs.fr
- researchgate.net
- math.stackexchange.com
- comptes-rendus.academie-sciences.fr
- semanticscholar.org
- www-cdn.anthropic.com
- anthropic.com
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
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