Gross's Euler–Poincaré orbital-integral conjecture

From papers

Let FF be a non-Archimedean local field, let GG be the reductive group under consideration, and let fEPf^{EP} be the Euler–Poincaré function with respect to a Haar measure dgdg on G(F)G(F). For a semisimple element γ\gamma with centralizer Gγ(F)G_\gamma(F), write Oγ(fEP,dg/dgγ)O_\gamma(f^{EP},dg/dg_\gamma) for its orbital integral, let dgγdg_\gamma be any Haar measure on Gγ(F)G_\gamma(F), and let dgγEPdg_\gamma^{EP} be the Euler–Poincaré measure on Gγ(F)G_\gamma(F). Gross's Euler–Poincaré conjecture.

Oγ(fEP,dg/dgγ)dgγ={dgγEPif γ is elliptic semisimple,0otherwise.O_\gamma(f^{EP},dg/dg_\gamma)dg_\gamma=\begin{cases}dg_\gamma^{EP}&\text{if }\gamma\text{ is elliptic semisimple},\\0&\text{otherwise.}\end{cases}

The conjecture identifies the orbital integrals of the Euler–Poincaré function with Euler–Poincaré measures precisely on elliptic semisimple classes. It is attributed here to Gross; the supplied text does not state whether it has been resolved.

Progress summary

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Sources & referencesView supporting material

Primary source

Takuro Fukayama, “The number of cuspidal representations over a function field and its behavior under base changes”, arXiv:2504.20564 (2025).

Additional references

2 papers in this index state this conjecture (2022–2025). The statement above is taken from the most recent of them; the others are arXiv:2208.01069.

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