Galois invariance of local root numbers
Using results of Fröhlich-Queyrut, Martinet, and Serre, it is shown that under appropriate hypothesis the local root number \(W(\rho,\psi)\) attached to a representation \(\rho\) of the Weil-Deligne group \(\text{WD}(\overline{F}/F)\) of a \(p\)-adic field \(F\) is invariant under Galois conjugation of the coefficients of the representation. In more detail, let \(\iota\) be any element of \(\Aut\mathbb C\). Proposition 4 asserts that if \(\rho\) is a representation of the Weil group \(W(\overline{F}/F)\), then: (i) If \(\rho\) is essentially self-dual and \(\det\rho\) is essentially trivial then \(W(\rho,\psi)\) is \(\pm 1\) and independent of \(\psi\). (ii) If \(\rho\) is essentially self-dual then so is the conjugate \(\rho^\iota\). (iii) If \(\rho\) is essentially self-dual and dim \(\rho\) is even, if det\(\rho\) is essentially trivial then so is det \(\rho^\iota\). Then (iv) If \(\rho\) is essentially self-dual, \(\dim\rho\) is even, and \(\det\rho\) is essentially trivial then \(W(\rho^\iota,\psi)\) is \(\pm 1\) and independent of \(\psi\). Assume that \(\rho\) is essentially self-dual and det\(\rho\) is essentially trivial. Then we can write \(W(\rho)\) for \(W(\rho,\psi)\). Theorem 1 asserts that if \(\rho\) is also of odd weight and even dimension, then \(W(\rho^\iota)=W(\rho)\). Three examples following Theorem 1 show that the requirements that \(\dim\rho\) be even, that \(\det\rho\) is essentially trivial, and that the weight is odd, cannot be omitted. Proposition 12 and Theorem 2 are generalizations to the context of Weil-Deligne groups. Of course \(\psi\) is a nontrivial unitary character of \(F\) into \(\mathbb C^\times\), and not into \(\mathbb C\) as written on page 987, line before \((2.3)\), but otherwise the exposition in this paper is exemplary.
scientific article; zbMATH DE number 5988221
scientific article