In the theory of automorphic representations one says that G satisfies a "multiplicity one" property if every cuspidal representation occurs with multiplicity one in $L^2(G(F)\backslash G(A))$.


One also says that G satisfies "strong multiplicity one" if a cuspidal representation is uniquely determined up to isomorphism by its behaviour at cofinitely many places of F. 


My question is: does the latter always imply the former (as the name would suggest), and is there a decent reason?


This question was prompted while thinking about Jacquet-Langlands, whose proof (via the trace formula) seems to give multiplicity one as a by-product and whose statement also directly implies strong multiplicity one (for inner forms of GL2 using the result for GL2 itself). However, I didn't see any way to deduce multiplicity one directly from the usual statement.


I also can't see how strong multiplicity one for classical modular forms should directly imply the q-expansion principle (which I guess is roughly the same as multiplicity one). Apologies if I'm missing something extremely obvious on this dozy Sunday evening.


Thanks,
Tom.



Is strong multiplicity one (obviously) stronger than multiplicity one?
I'm not sufficiently privileged to leave comments yet, but in support of the semantics I'll mentioned the following paper:
<http://www2.math.ou.edu/~rschmidt/papers/sqfree.pdf>


In this paper, Schmidt refers to his Corollaries 3.16 and 3.17 as "strong multiplicity one theorems". This would agree with Tom's definition of "strong multiplicity one". In the case of $GSp(4)$ "multiplicity one" is not known (representations do not necessarily have Whittaker models, which complicates things), so even though Schmidt can say isomorphic at cofintely many place implies isomorphic, he is not able to say that the global representations are literally equal because the latter may (but are not expected to) occur with multiplicity greater than one in $L^2(GSp(4, \mathbb{Q}) \backslash GSp(4, \mathbb{A})$.


So, from an appropriate point of view, Bump's statement of Theorem 3.3.6 is a "multiplicity one" + "strong multiplicity one" (the former of which is known to be true for $GL\_2$ of course).


