Frobenius Equidistribution and Arithmetic of Fibers in Algebraic Families

Tianhao Wang

University of California, Irvine

May 14th, 2026

The Frobenius at a Closed Point
  • $X$: a geometrically integral scheme of finite type over $\mathbb{Z}$ or $\mathbb{F}_q$.
  • $x_0\in |X|$: a closed point with finite residue field $\kappa(x_0)$.
  • $\overline{\eta}$: a geometric generic point of $X$.
  • Choose a specialization $\overline{\eta}\leadsto \overline{x}_0$, where $\overline{x}_0$ is a geometric point above $x_0$. The morphism $\overline{x}_0\to X$, together with this specialization, induces a homomorphism, well-defined up to conjugation, $$ D_{x_0}: \operatorname{Gal}\bigl(\overline{\kappa(x_0)}/\kappa(x_0)\bigr) \longrightarrow \pi_1^{\mathrm{et}}(X,\overline{\eta}). $$ The image of $a\mapsto a^{|\kappa(x_0)|}$ under $D_{x_0}$ defines the Frobenius conjugacy class.
Poncelet $n$-gon and Poncelet Porism
Outer ellipse A (blue)
Inner ellipse B (red)
Involutions \(i_1,i_2:E\to E\) and composition \(j=i_2\circ i_1\): \[i_1:(P,\xi)\mapsto (P',\xi), \qquad i_2:(P',\xi)\mapsto (P',\xi')\] Poncelet \(n\)-gon: \(j^n(P_0,\xi_0)=(P_0,\xi_0)\).
If \((\mathcal{A},\mathcal{B})\) admits a Poncelet \(n\)-gon for some starting point \((P_0, \xi_0)\), then it admits one for every starting point on \(\mathcal{A}\).
Poncelet’s Porism & Griffiths–Harris Theorem
Outer ellipse A (blue)
Inner ellipse B (red)
Thank You

Tianhao Wang

University of California, Irvine

tianhw11@uci.edu

Frobenius Equidistribution and Arithmetic of Fibers in Algebraic Families