Hello,
I began to read a few weeks ago an article about automorphic L-functions in which a formula like $L(s,\pi\times\pi')=L(s,\pi)L(s,\pi')$ appeared. Unfortunately, I can't find it back. Could someone give me some reference? Thank you in advance.
$\begingroup$
$\endgroup$
1
asked Jul 20, 2011 at 5:21
-
6$\begingroup$ The identity $L(s,\pi \boxplus \pi')=L(s,\pi)L(s,\pi')$ is true, where $\boxplus$ indicates "isobaric sum" of automorphic representations (part of Langlands's theory of Eisenstein series). $\endgroup$– David HansenCommented Jul 20, 2011 at 5:43
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
4
As David Hansen says, Langlands proved that there is an automorphic representation (consisting of specific Eisenstein series) whose $L$-function is your right hand side. This representation is denoted by $\pi \boxplus \pi'$ and is called the isobaric sum of $\pi$ and $\pi'$, it mimics the direct sum of Galois representations. The notation $\pi\times\pi'$ or $\pi\otimes\pi'$ usually denotes the Rankin-Selberg convolution: it mimics the tensor product of Galois representations, and its $L$-function is not your right hand side.
answered Jul 20, 2011 at 8:32
-
1$\begingroup$ Thank you very much. Indeed the notation that I used is incorrect, but I just couldn't remember the right one. And, as a matter of fact, that is why I wrote "like". Anyway it shows that the product of any two automorphic L-functions is again an automorphic L-function, am I wrong? $\endgroup$ Commented Jul 20, 2011 at 17:34
-
$\begingroup$ Yes, the product of two automorphic $L$-functions is again automorphic. Moreover, any automorphic $L$-function has a unique maximal factorization into automorphic $L$-functions, and the factors correspond to cusp forms on possibly smaller groups. Actually, here I have automorphic representations of $\mathrm{GL}_n$ in mind (which produce the so-called principal $L$-functions), I don't know much about the general theory. $\endgroup$ Commented Jul 20, 2011 at 18:07
-
$\begingroup$ @paul: I will now delete my comment to your comment, so that you can delete your comment to my comment to your comment :) $\endgroup$ Commented Jul 20, 2011 at 21:14
-
2$\begingroup$ But another point: sometimes or often it is implicit that one is interested in whether a product L-function (from two cuspidal) is again itself cuspidal (as opposed to being "of course" attached to suitable Eisenstein series). This "cuspidal?" version of the question is much harder, but/and whenever answered affirmatively gives bounds, entireness, and such stuff that do not follow for L-functions attached to not-necessarily-cuspidal data. $\endgroup$ Commented Jul 20, 2011 at 23:20