Ilya Piatetski-Shapiro

Quadratic base change of θ10

In case ofGLn overp-adic fields, it is known that Shintani base change is well behaved. However, things are not so simple for general reductive groups. In the first part of this paper, we present a counterexample to the existence of quadratic base change descent for some Galois invariant representations. These are representations of type θ10. In the second part, we compute the localL-factor of θ10. Unlike many other supercuspidal representations, we find that theL-factor of θ10 has two poles. Finally, we discuss these two results in relation to the local Langlands correspondence.

STABILITY OF

One of the main obstacles in applying converse theorems to prove new cases of functoriality is that of stability of

\gamma

\gamma

-FACTORS FOR QUASI-SPLIT GROUPS

-factors for a certain class of

Base change for the Saito-Kurokawa representations of PGSp(4)

L

Derivatives and L-Functions for GL n

-functions obtained from the ‘Langlands–Shahidi’ method, where the

Unitarity and Functoriality.

\gamma

Base change for ≈SL2

-factors are defined inductively by means of ‘local coefficients’. The problem then becomes that of stability of local coefficients upon twisting the representation by a highly ramified character. In this paper we first establish that the inverses of certain local coefficients are, up to an abelian

On base change for odd orthogonal groups

\gamma

Functoriality for the classical groups

-factor, genuine Mellin transforms of partial Bessel functions of the type we analysed in our previous paper. The second main result is then the resulting stability of the local coefficients in this situation, which include all the cases of interest for functoriality. Hopefully, the analysis given here will open the door to a proof of the general stability and the equality of