Inductivity of the Global Root Number

The invariance of the Artin root number under induction can be proved without special effort. In fact if one develops the properties of Artin Lfunctions in the usual way, then the inductivity of the root number is obvious. In barest outline the argument is as follows. First one proves the inductivity of Artin L-functions themselves, and then one combines Brauer’s induction theorem with the analytic continuation and functional equation of Hecke L-functions, thereby deriving the analytic continuation and functional equation of Artin L-functions. The inductivity of the gamma and exponential factors follows from the duplication formula and the properties of the Artin conductor respectively, so the “normalized L-function” (the product of the L-function and its exponential and gamma factors) is also inductive. Since the root number is the normalized L-function at s divided by the normalized dual L-function at 1−s, the inductivity of the root number follows. Artin L-functions can be viewed as the simplest examples of motivic L-functions, but in the general setting much less is known. Certainly the analytic continuation and functional equation are only conjectural, and even the basic objects—the L-function, conductor, and root number—are conjectural in the sense that their definition depends on compatibilities which are not yet known to hold in complete generality. Paradoxically, these very limitations on our knowledge make it easy to prove that motivic root numbers are inductive: Given that the root number itself is only conjectural, one is happy to use the conjectural functional equation, and then an argument along the lines of the proof sketched above goes through as before. The present note deals with an intermediate case. We consider a setting in which the analytic continuation and functional equation of the L-function are in general still conjectural, while the L-function, conductor, and root number are well defined. The example to keep in mind is an elliptic curve over an arbitrary number field, where we know the required compatibili-