[Paper Review] Langlands program for $p$-adic coefficients and the petites camarades conjecture

Tomoyuki Abe|arXiv (Cornell University)|Nov 10, 2011

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes a Langlands-type correspondence for $p$-adic coefficients on smooth curves over finite fields, proving that Deligne's 'petites camarades' conjecture implies the Langlands correspondence for $p$-adic representations. It further shows that all overconvergent $F$-isocrystals of rank $\leq 2$ on smooth varieties are $\iota$-mixed, supporting a key part of the $p$-adic Langlands program.

ABSTRACT

In this paper, we prove that, if Deligne's "petites camarades conjecture" holds, then a Langlands type correspondence holds also for $p$-adic coefficients on a smooth curve over a finite field. We also prove that any overconvergent $F$-isocrystal of rank less than or equal to 2 on a smooth variety is $ι$-mixed.

Motivation & Objective

• To establish a Langlands-type correspondence for $p$-adic coefficients on smooth curves over finite fields.
• To prove the equivalence between the Langlands program for $p$-adic coefficients and Deligne's petites camarades conjecture.
• To show that all overconvergent $F$-isocrystals of rank $\leq 2$ on smooth varieties are $\iota$-mixed.
• To extend the Langlands correspondence to $p$-adic coefficients using $p$-adic epsilon factors and reciprocity maps.

Proposed method

• Uses the product formula for $p$-adic $\varepsilon$-factors from [AM] to deduce the Langlands correspondence from the petites camarades conjecture.
• Applies standard arguments from [De1] and [Lau] to establish the correspondence via matching Frobenius and Hecke eigenvalues at unramified places.
• Employs the reciprocity map and Tsuzuki's result to construct the Langlands correspondence for rank 1 $F$-isocrystals.
• Reduces the $\iota$-mixedness of higher rank $F$-isocrystals to the rank 1 and 2 cases using base change and restriction to curves.
• Uses the notion of $\iota$-mixedness via stratifications and specialization of overconvergent $F$-isocrystals to prove the main result on rank $\leq 2$.
• Applies the theory of overconvergent $F$-isocrystals on $d$-varieties and overholonomic $F$-$\mathscr{D}^\dagger$-complexes to generalize the result.

Experimental results

Research questions

• RQ1 Does the Langlands correspondence for $p$-adic coefficients hold under the assumption of Deligne's petites camarades conjecture?
• RQ2 Are all overconvergent $F$-isocrystals of rank $\leq 2$ on smooth varieties $\iota$-mixed?
• RQ3 Can the Langlands correspondence be extended from $\ell$-adic to $p$-adic coefficients using epsilon factor identities?
• RQ4 Is there a canonical correspondence between irreducible overconvergent $F$-isocrystals and automorphic representations for $\mathrm{GL}_r$ over function fields?

Key findings

• The Langlands correspondence for $p$-adic coefficients is equivalent to Deligne's petites camarades conjecture.
• The Langlands correspondence is established for rank 1 $F$-isocrystals via the reciprocity map and unit-root property.
• For rank 2, the correspondence is constructed using the coincidence of $\varepsilon$-factors and Claim 5.2.
• All overconvergent $F$-isocrystals of rank $\leq 2$ on smooth varieties are $\iota$-mixed, as shown via reduction to curves and base change.
• The $\iota$-mixedness of overholonomic $F$-$\mathscr{D}^\dagger$-complexes on $d$-varieties follows from the conjecture and the rank $\leq 2$ result.
• The proof relies on the product formula for $p$-adic $\varepsilon$-factors and standard techniques from $\ell$-adic Langlands theory.