Local orders, root numbers, and modular curves. (English) Zbl 0675.12011

Gross, Benedict H.

Am. J. Math. 110, No. 6, 1153-1182 (1988).

Let F be a local field. Let \(\pi\) be an irreducible admissible representation of \(GL(2,F)\) and \(\pi_{\chi}\) the representation of \(GL(2,F)\) which corresponds (via the Weil representation) to a character \(\chi\) of a quadratic extension of F. Assume the restriction of \(\chi^{-1}\) to \(F^*\) is equal to the central character of \(\pi\). Then \(\pi \times \pi_{\chi}\) is isomorphic to its dual and its L-function gives us a root number \(\pm 1.\) Using the determination of this root number the author associates to \(\pi\) and \(\chi\) a quaternion algebra B over F and an irreducible representation E of \(B^*\), which is either \(\pi\) or the representation corresponding to it according to Jacquet- Langlands in the case when B is the division algebra. When \(\pi\) and \(\pi_{\chi}\) have little common ramification a certain order R in B is defined such that E contains (up to a scalar) a unique vector invariant under \(R^*\). This is a generalization of the theory of new vectors.
This local result is applied in the global situation to a certain eigenspace of the Hecke algebra. Here the two different cases are considered: the local algebras \(B_ v\) may come from a global quaternion algebra or not.

Reviewer: J.G.M.Mars

MSC:

11S45	Algebras and orders, and their zeta functions
11F70	Representation-theoretic methods; automorphic representations over local and global fields
11S40	Zeta functions and \(L\)-functions
11S37	Langlands-Weil conjectures, nonabelian class field theory
20G25	Linear algebraic groups over local fields and their integers
14H25	Arithmetic ground fields for curves

Keywords:

L-function; root number; new vectors; eigenspace of the Hecke algebra