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Local geometrized Rankin-Selberg method for \(\text{GL}(n\)). (English) Zbl 1080.11040

Let \(F\) be the function field of a curve \(X\) over a finite field \(k\). Automorphic forms on \(\text{GL}(n,F_{{\mathbb A}})\) can, when they belong to an everywhere unramified representation, be considered as functions on the set of isomorphism classes of vector bundles of rank \(n\) on \(X\). Let \(\phi_E\) be the cuspidal automorphic form associated by the Langlands correspondence for \(\text{GL}(n)\) to an irreducible local system \(E\) of rank \(n\) on \(X\) (i.e. a geometrically irreducible smooth \(\overline{{\mathbb Q}}_l\)-sheaf). \(\phi_E\) is a Hecke eigenvector with respect to \(E\). In this situation, the Rankin-Selberg method, which is a method to study automorphic \(L\)-functions for \(\text{GL}(n) \times \text{GL}(n)\), consists of the computation for any \(d\) of the scalar product of \(\phi_{E_1}\) and \(\phi_{E_2}\) on the set of isomorphism classes of vector bundles of rank \(n\) and degree \(d\) on \(X\).
Independently of the Langlands correspondence, one can define the restriction \(\widetilde{\phi}_E\) of \(\phi_E\) on the set of isomorphism classes of sheaf imbeddings \(\Omega^{n-1} \rightarrow L\), where \(L\) is a vector bundle of rank \(n\) on \(X\) and \(\Omega\) is the canonical invertible sheaf on \(X\). Now the global result can be derived from a local formula, which is an equality of formal series, namely, the sum over \(d\geq 0\) of \(t^d\) times the scalar product of \(\widetilde{\phi}_{E_1}\) and \(\widetilde{\phi}_{E_2}\) on the set of isomorphism classes of \((\Omega^{n-1} \rightarrow L)\) with \(\text{degree}(L)=d+n(n-1)(g-1)\) is equal to the \(L\)-function of \(E_1 \otimes E_2\).
A geometric version of this equality is proved in the paper. The base field \(k\) is assumed to be algebraically closed. As a geometric counterpart of \(\widetilde{\phi}_E\) we have a complex \({\mathcal K}_E\) of \(l\)-adic sheaves on the moduli stack of the imbeddings \(\Omega^{n-1} \rightarrow L\), defined by G. Laumon [Duke Math. J. 54, No. 2, 309–359 (1987; Zbl 0662.12013)]. The result is an expression for the cohomology of \({\mathcal K}_{E_1} \boxtimes {\mathcal K}_{E_2}\).

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11R39 Langlands-Weil conjectures, nonabelian class field theory
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 0662.12013
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