1 results
RANK 2
$\ell $-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu , First View
- Published online by Cambridge University Press:
- 10 February 2025, pp. 1-70
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
Let X be a smooth, projective and geometrically connected curve defined over a finite field
${\mathbb {F}}_q$ of characteristic p different from 
$2$ and 
$S\subseteq X$ a subset of closed points. Let 
$\overline {X}$ and 
$\overline {S}$ be their base changes to an algebraic closure of 
${\mathbb {F}}_q$. We study the number of 
$\ell $-adic local systems 
$(\ell \neq p)$ in rank 
$2$ over 
$\overline {X}-\overline {S}$ with all possible prescribed tame local monodromies fixed by k-fold iterated action of Frobenius endomorphism for every 
$k\geq 1$. In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.
