The semi-global arithmetic intersection formula

Fix a place $v_0$ of $F_0$ above a place $p\leq\infty$ of $\mathbb Q$. Let $f$ be a completely decomposed element of the relevant partial Hecke algebra and let $f'=\otimes_v f'_v$ be a Gaussian test function on $G'=\operatorname{Res}_{F/F_0}(\mathrm{GL}_{n-1}\times\mathrm{GL}_n)$ whose finite part is a smooth transfer of $f$. Assume regular support at a place $\lambda$ away from $p$.

Semi-global arithmetic intersection conjecture. If $v_0$ is non-archimedean of hyperspecial type and $f'_{v_0}=\mathbf 1_{G'(O_{F_0,v_0})}$, then

$$\operatorname{Int}_{v_0}(f)=-\partial J_{v_0}(f').$$

If $v_0$ is archimedean or non-archimedean of AT type, then

$$\operatorname{Int}_{v_0}(f)=-\partial J_{v_0}(f')-J(f'_{\mathrm{corr}}[v_0]),$$

where $f'_{\mathrm{corr}}[v_0]$ is a correction function, and $f'$ may be chosen so that this correction function is zero.

This is a local-at-$v_0$ refinement of the global arithmetic intersection formula and is presented as conjectural, with concrete evidence available in the semi-global setting.