The Tate-linear conjecture for products of GSpin Shimura varieties

About 3 years old · traced to 2023

Let $f:X_0\to\mathscr{S}_{\mathbf{I},F_{\infty}}^{\operatorname{ord}}$ be a locally closed immersion, so that $X$ is a subvariety of $\mathscr{S}_{\mathbf{I},F_{\infty}}^{\operatorname{ord}}$ , and let $x\in X(F_{\infty})$ . Say that $X$ is Tate-linear at $x$ when its formal completion at $x$ has the Tate-linear property. The Tate-linear conjecture for products of GSpin Shimura varieties. If $X$ is Tate-linear at $x$ , then $X$ is special.

This is the explicit geometric formulation of the product-GSpin Tate-linear conjecture. The supplied passage gives no resolution status for it.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Tate-linear conjecture for products of GSpin Shimura varieties
Let $(X,x)$ , $X_0$ , $f$ , and $\T_{f,x}$ be as in the source, with $f:X_0\to X$ and with $\T_{f,x}$ the associated Tate-linear formal object. Let $\X_{f,F_{\infty},\operatorname{red}}^{/x}$ denote the formal reduction associated with $f$ at $x$ . The Tate-linear conjecture for products of GSpin Shimura varieties. One should have
$\mathscr{X}_{f,\mathbb{F},\operatorname{red}}^{/x}=\mathscr{T}_{f,x}.$
The equality is the product-GSpin formulation of Tate-linearity used in the paper. The supplied passage does not give a resolution status for this particular formulation.
source: Ruofan Jiang, “p-adic monodromy and mod p unlikely intersections, I”, arXiv:2308.06854 (2025).

References

Primary source

Ruofan Jiang, “p-adic monodromy and mod p unlikely intersections, I”, arXiv:2308.06854 (2025).

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