Partial Frobenius on shtukas

For a multiple-leg $G$ -shtuka with legs indexed by $I$ , a partial Frobenius associated to a subset $J\subset I$ applies the $q$ -power Frobenius endomorphism to the leg coordinates in $J$ while leaving the other legs fixed, together with the corresponding rotation of the Frobenius-modification chain.

On the base $X^I$ , the map is

{\rm Frob}_J((x_i)_{i\in I})=(x_i')_{i\in I}, \qquad x_i'= \begin{cases} {\rm Frob}_X(x_i),&i\in J,\\ x_i,&i\notin J. \end{cases}

Modification-chain description

For an ordered partition $I=I_1\sqcup\cdots\sqcup I_k$ , a shtuka is a chain from $\mathcal G_0$ to ${}^\tau\mathcal G_0$ . Partial Frobenius for the first block moves that block to the end, replaces $\mathcal G_0$ by the next bundle in the chain, and applies Frobenius to the moved legs. This gives a morphism between the appropriately ordered shtuka stacks.

Relations

Partial Frobenius operators for disjoint individual legs commute. Their product over all legs is the total Frobenius action. On the inductive system of truncated shtuka cohomology, an operator may enlarge the Harder–Narasimhan bound; it is therefore naturally a map in that inductive system rather than always an endomorphism of one finite-type truncation.

From Frobenius to Galois

Drinfeld's lemma turns a lisse $\ell$ -adic sheaf over a power of a curve equipped with commuting partial Frobenius structures into a representation of a product of fundamental groups. This is the source of the independent Galois actions $(\gamma_i)_{i\in I}$ in an excursion operator.

References

Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” §§0.2–0.4 and Chapter 3. arXiv.
Cong Xue, “Smoothness of cohomology sheaves of stacks of shtukas,” 2020. arXiv.