University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > A uniform open image theorem for $ll$-adic representations (joint work with Akio Tamagawa - R.I.M.S.)

A uniform open image theorem for $ll$-adic representations (joint work with Akio Tamagawa - R.I.M.S.)

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Non-Abelian Fundamental Groups in Arithmetic Geometry

In this talk, we extend some of the results presented in Tamagawa’s talk to more general $ll$-adic representations.\ indent Let $k$ be a finitely generated field of characteristic $0$, $X$ a smooth, separated, geometrically connected curve over $k$ with generic point $ta$. A $ll$-adic representation $ ho:pi_{1}(X) ightarrow hbox{ m GL}(mathbb{Z}{ll})$ is said to be geometrically strictly rationnally perfect (GSRP for short) if $hbox{ m Lie}( ho(pi_{1}(X_{overline{k}})))=0$. Typical examples of such representations are those arising from the action of $pi_{1}(X)$ on the generic $ll$-adic Tate module $T_{ll}(A_{ta})$ of an abelian scheme $A$ over $X$ or, more generally, from the action of $pi_{1}(X)$ on the $ll$-adic etale cohomology groups $H{i}(Y_{overline{ta}},mathbb{Q}_{ll})$, $igeq 0$ of the geometric generic fiber of a smooth proper scheme $Y$ over $X$. Let $G$ denote the image of $ ho$. Any closed point $x$ on $X$ induces a splitting $x:Gamma_{kappa(x)}:=pi_{1}(hbox{ m Spec}(kappa(x)))

ightarrowpi_{1}(X_{kappa(x)})$ of the canonical restriction epimorphism $pi_{1}(X_{kappa(x)}) ightarrow Gamma_{kappa(x)}$ (here, $kappa(x)$ denotes the field of definition of $x$) so one can define the closed subgroup $G_{x}:= ho rc x(Gamma_{kappa(x)}) ubset G$ (up to inner automorphisms).\ indent The main result I am going to discuss is the following uniform open image theorem. extit{Under the above assumptions, for any representation $ ho:pi_{1}(X) ightarrow hbox{ m GL}(mathbb{Z}{ll})$ and any integer $dgeq 1$, the set $X_{ ho, d,geq 3}$ of all closed points $xin X$ such that $G_{x}$ has codimension $geq 3$ in $G$ and $[kappa(x):k]leq d$ is finite. Furthermore, if $ ho:pi_{1}(X) ightarrow hbox{ m GL}(mathbb{Z}{ll})$ is GSRP then the set $X_{ ho, d,geq 1}$ of all closed points $xin X$ such that $G_{x}$ has codimension $geq 1$ in $G$ and $[kappa(x):k]leq d$ is finite and there exists an integer $B_{ ho,d}geq 1$ such that $[G:G_{x}]leq B_{ ho,d}$ for any closed point $xin X mallsetminus X_{ ho,d,geq 1}$ such that $[kappa(x):k]leq d$.}\

This talk is part of the Isaac Newton Institute Seminar Series series.

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