Let X/k be a smooth variety over a finite field. Motivated by work of Corlette-Simpson over the complex numbers, we formulate a conjecture that certain rank 2 local systems on X come from families of abelian varieties. After an introduction to l/p-adic companions, we explain how the existence of p-adic companions can be used to approach the conjecture via a Lefschetz-style theorem for families of abelian varieties in positive characteristic. This is joint work with A. Pál.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3207