Yifan Wei

Progress to date: A Chebotarev Density Theorem over Local Fields with Asvin G and John Yin

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Oct. 23, 2025

Let $\pi \colon X \to Y$ be a generically Galois map between p-adic schemes, we can comute the measure $\mu(p)$ of the image $\pi(X(\mathbb{Z}_p))$ in $Y(\mathbb{Z}_p)$ using a p-adic measure on $Y(\mathbb{Z}_p)$. In the situation when $Y$ is smooth projective over $\mathbb{Z}_p$, and a form of resolution of singularity for $\pi$ holds, the measure $\mu(p)$ satisfies a remarkable symmetry with regard to the transformation $p \to 1/p$ (vaguely speaking). The reason for this is two fold, one is that the density for some local maps like $\mathbb{P}^1 \stackrel{n}{\to} \mathbb{P}^1$ satisfies nice symmetries, another is that resolution of singularity provides a nice combinatorial situation so the symmetry is preserved.

Recently I'm thinking about this problem again, and it seems the density for isomorphism types of integral Galois representations might also satisfy certain symmetry properties. For example we may take the modular curve $Y(1)$ to be our p-adic base, and take the integral $\ell$-adic Tate module of associated elliptic curves to be the integral local system $\mathcal{L}$. Let $x_0 \in Y(1)(\mathbb{Q}_p)$ be a fixed base point, we can compute the measure of the following set $\{x \in Y(1)(\mathbb{Q}_p)\ |\ \mathcal{L}_x \cong \mathcal{L}_{x_0} \}$, where the disk at infinity on $Y(1)$ is normalized to have measure one. It turns out the density again has this $p \to 1/p$ symmetry (regardless of the valuation of $x_0$), although the exact density is dependent on $\ell$. Similar symmetry might exist for the moduli space of higher dimensional abelian varieties, but I have not computed these examples yet.

(It can be argued that we haven't found the most natural way to pose this question, since having an $\ell$ floating around doesn't seem very natural.)

Progress to date: Matrix Points on Varieties with Asvin G, Yifeng Huang and Ruofan Jiang.

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Oct. 23, 2025

A matrix point on an affine $k$-variety $X = \mathrm{Spec} R$ is a $k$-algebra homomorphism $R \to \mathrm{M}_n(k)$. Let's denote the set of matrix points on $X$ by $X(\mathrm{M}_n(k))$. In https://arxiv.org/abs/2110.15566 Yifeng Huang wrote down the following beautiful sum-product formulae for the $\mathrm{GL}_n(\mathbb{F}_q)$-weighted generating series of $X(\mathrm{M}_n(\mathbb{F}_q))$:

What's the cohomological reason for the above product factorization? We have been thinking about this question since early 2022 and our understanding keeps evolving. Currently we are able to understand the rational cohomology of the space of matrix points. We denote the moduli space of matrix points on $X$ by $C_n(X)$, which is seen as a non-commutative Weil restriction from the algebra of $n$-by-$n$ matrices to the ground field.

The rational cohomology of $C_n(X)$ also fits in the Weil restriction heuristics: we may think of $k \to \mathrm{M}_n(k)$ as a non-commutative version of a separable degree $n$ "field extension". Let's pretend that $S_n$ is the Galois group of this field extension, and that there exists a Galois closure $\hat{\mathrm{M}}_n(k)$ of the extension $k \to \mathrm{M}_n(k)$. Then from the classical theory of Weil restrictions, we expect $$C_n(X) = X^n \times^{S_n} ``\mathrm{Spec} \hat{\mathrm{M}}_n(k)".$$

Our current result says that cohomologically, $C_n(X)$ behaves as $X^n \times^{S_n} \mathrm{GL}_n/\mathrm{T}_n$, where $H^*(\mathrm{GL}_n/\mathrm{T}_n)$ serves as the cohomological model for the "Galois closure" of the algebra $\mathrm{M}_n$. As a consequence, we deduced a rather elegant combinatorial formula for the rational cohomology of $C_n(X)$.

The Poincare polynomial of $C_n(X)$ can be computed using principal specialization of $S_n$ characters on the cohomology of $X^n$:

$$P_u(C_n(X)) = \phi_n(u^2) \mathrm{sp}_{u^2}(\mathbf{ch}_u(H^*(X^n)))$$

where $\mathrm{sp}_{u^2}$ is the principal specialization, $\phi_n(u^2) = \prod_i^n (1 - u^{2i})$, and $\mathbf{ch}_u$ is the graded $S_n$-character. Since the $S_n$-characters of $H^*(X^n)$ form a nice generating series as $n$ varies --- there is a sum-product formula for this generating series, we obtain a sum-product formula for the generating series of the Poincare polynomial of $C_n(X)$ as well. This is a geometric-combinatorial explanation for the sum-product formula mentioned above in the case of smooth curves, since $C_n(X)$ is smooth in that case, the point count is completely determined by the Frobenius trace on its cohomologies.

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Oct. 23, 2025

Inspired by Asvin G's paper https://arxiv.org/abs/2203.16774 where the eigenvalues of Frobenius satisfies an $\ell$-adic convergence on an $\ell$-adic tower, we proceeded to understand more examples where we see some king of $\ell$-adic convergence.

We focused on understanding nilpotent $\ell$-adic towers of $X = \mathbb{P}^1 - \{0, 1, \infty\}$. Let $\hat{\pi}$ be the geometric fundamental group (with tangential base point at $0$), it is isomorphic to the profinite completion of the free group on two generators. The pro-$\ell$ abelianization of $\hat{\pi}$ produces a tower of geometrically Galois covers $X_n \to X$ (with non-constant abelian Galois group), the same technique as in the original Asvin G's paper shows that the Frobenius eigenvalues on $X_n$ satisfies an $\ell$-adic convergence property. We also computed the pro-$\ell$ Heisenberg tower (maximal class two pro-$\ell$ nilpotent quotient of $\hat{\pi}$), and the Frobenius eigenvalues again satisfies some $\ell$-adic convergence, albeit with some normalization involved.

What really went into the computation turns out to be purely group theoretic, and we are still not certain what's really happening. What's a deeper reason that we must see these (normalized) convergences?