Xxxx’s Conjecture Sample Clauses

Xxxx’s Conjecture. (Conjecture 1.5) is valid in both of the following cases: (a) [See Theorem 7.7] L/K is tamely ramified and a mild hypothesis on the unramified characteristics of ramified places (which is made precise in the statement of Theorem 7.7) is satisfied. (b) [See Theorem 7.11] The conditions of claim (i)(b) and the hypothesis of claim(ii)(a) on the unramified characteristics of ramified places are satisfied and, in addition, the inertia degree of each wildly ramified place v in L/K is prime to the absolute degree of the completion Kv. We remark that claim (ii)(a) of this result removes the locally abelian hypothesis from the main result of Xxxxxx and Vinatier in [13]. We also note that, whilst the result of claim (ii)(a) was first proved in the jointly authored article [2], the argument that we present here closely follows the classical methods of Erez in [20] (see Remark 7.15(i)) rather than the methods of Xxxxxxx and XxXxxxxx in [3] and so is different in key respects from that given in [2]. We further note that it seems possible that the technical condition on unramified characteristics that occurs in claim (ii)(a) is not always satisfied and hence that this result could actually lead to a counterexample to the conjecture of Erez (for more details see Remark 7.9). Finally, we note that the result of claim (i) of Theorem 1.6 makes it natural to wonder whether, with our generalized definitions of the elements aL/K and cL/K, the equality aL/K = cL/K that is conjectured by Xxxx, Xxxxx and Xxxx should be valid for all weakly ramified Galois extensions L/K for which the fractional ideal AL/K exists. This is a question that we feel deserves further careful attention. However, since we have no evidence for an affirmative answer to it that goes beyond that in Theorem 1.6 and, in particular, we lack the sort of numerical evidence that is provided in [5], we shall end this introduction by simply posing the following question.