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**On the direct image conjecture in the relative Langlands programme.**
*(English.
Russian original)*
Zbl 1371.14019

Russ. Math. Surv. 70, No. 5, 961-963 (2015); translation from Usp. Mat. Nauk 70, No. 5, 181-182 (2015).

In an earlier paper [Russ. Math. Surv. 67, No. 3, 509–539 (2012; Zbl 1318.22010)], the author discussed the “direct image conjecture.” This conjecture asserts that a direct image operation exists and has certain properties, in the case of a proper flat morphism \(f: X\to B\) with smooth generic fiber, where \(X\) is a surface and \(B\) is a regular curve. He also outlined a proof of this conjecture in the case in which \(f\) is smooth and \(X\) and \(B\) are smooth and proper over a finite field. This proof relies on the Langlands correspondence for function fields of one variable established by Laurent Lafforgue.

The paper under review extends this result by relaxing the condition that the morphism be smooth. Instead, \(f\) is required to have smooth generic fiber and its degenerate fibers may have as singularities only rational double points. Some additional properties are also shown to hold.

The paper under review extends this result by relaxing the condition that the morphism be smooth. Instead, \(f\) is required to have smooth generic fiber and its degenerate fibers may have as singularities only rational double points. Some additional properties are also shown to hold.

Reviewer: Paul Vojta (Berkeley)

### MSC:

14D24 | Geometric Langlands program (algebro-geometric aspects) |

22E57 | Geometric Langlands program: representation-theoretic aspects |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |