Trace on Cp

On the Existence of Trace for Elements of % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX % garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz % aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaacq % WIceYOdaWgaaWcbaGaamiCaaqabaaaaa!3A08! $$\mathbb{C}_p$$

“…OðxÞ which sends ' 2 G to 'ðxÞ; is continuous and defines a homeomorphism between G=HðxÞ and OðxÞ; where HðxÞ is the stabilizer of x, see [1]. In such a way, OðxÞ can be endowed with a Haar measure, see [2]. This Haar measure has real values, which are rational numbers, and one can see them as being p<adic numbers.…”

AN ALGEBRAIC-METRIC EQUIVALENCE RELATION OVER p-ADIC FIELDS

“…This equivalence relation is defined in terms of field extensions of K and metric properties of Galois orbits over K. Our starting point is to consider the following natural question: Given two elements T and U of C p , under which circumstances is there a canonical way to define a map from the Galois orbit C K (T) to the Galois orbit C K (U)? The existence of such a map would have useful implications in questions related to integration along Galois orbits with respect to the Haar distribution, and to the problem of the existence of trace over K of transcendental elements over K (see [3], [5] and [11]). Given two elements T and U of C p , the obvious choice for the definition of a natural map from C K (T) to C K (U) would be to take each element z in C K (T), write it in the form z = σ (T) with σ ∈ G K and then send it to the element σ (U) of C K (U).…”

LIPSCHITZIAN ELEMENTS OVER p-ADIC FIELDS

“…As explained in [3], this condition is very useful in integration theory along Galois orbits in C p . Any Lipschitzian function f : C K (T) → C p is integrable with respect to the Haar distribution π K,T provided that T is Lipschitzian over K, even if π K,T is not bounded.…”

“…A larger class of elements T was defined in [3], in terms of a metric condition. For any real number > 0, the Galois orbit C K (T) of T can be written as a finite disjoint union of open balls of radius .…”