Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am wondering how the relation is between of the automorphic L-function and its lift's.

More precisely,

Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over E of dimension $n$, and let $V$ be a skew-hermitian space of dimension $m$ over $E$. Consider the associated unitary groups $H:=U(W)$ and $G:=U(V)$. Let $σ$ be an irreducible, cuspidal, automorphic representation of $H(A_F)$. Let $π=Θ(σ,ψ,γ)$ be a theta lift of $σ$ to $G(A_F)$. ($ψ:A_{F}/F→C^×$ and $γ:A_E^\times/E^{\times}→C^{\times}$ are the splitting data necessary to define the theta-lift for unitary groups.)

When $|m-n|\le1$ , it is conjectured and for small dimension case was discussed in chap 7 of
http://www.math.nus.edu.sg/~matgwt/ggp-goa-1-1.pdf

In the paper, it conjectures that the L-parameter of $\theta(\sigma)$, denoted by M, is tied to N, the L-parameter of $\sigma$ by $M=N$ for $m=n$ and $M=\mu^{-1} N \oplus \mu^{m}$ for $m=n-1$

But my primary concern lies in the case $m=n+2$, especially $n=1$ and $m=3$ or $n=2$ and $m=4$

Do you know the general rule for this case $m=n+2$ or even in the above special cases? If you know the answer or some reference regarding this and would you let me know? I will be very happy and grateful for your warm heart.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.