ɛ-Constants and the Galois Structure of de Rham Cohomology

Epsilon constants and orthogonal representations

“…Proof: For all places v ∈ S, this follows directly from [CEPT1]. However, one can generalize their results in order to prove the lemma.…”

“…We can choose such a D ′ after a suitable base change due to the moving lemma proven as Proposition 9.1.3 in [CEPT1]. The choice of this canonical divisor is not unique, but our calculation will show that the results are independent of the choice of D ′ .…”

“…The tameness hypotheses implies that the order of I is relatively prime to v, and that I is a cyclic group. The specific structure of I is discussed in detail in the Appendix to [CEPT1].…”

“…First, we must calculate terms ǫ(z, V ), where z ∈ Y is a closed point defined over a finite field. Section 2.5 of [CEPT1] gives us the following way of making this computation.…”

“…In particular, the definition of ǫ v,0 (Y, V ) requires that one chooses an auxiliary prime ℓ = v, a nontrivial continuous complex character of Q v which we denote by ψ v and a Haar measure dx v on Q v . In the case where V has trivial determinant and is of dimension 0, then ǫ v,0 (Y, V ) is independent of these choices (see Proposition 2.4.1 of [CEPT1]). This term is well-defined for v = ∞ as well as for finite places v.…”