On Saturated distinguished chairs over a local field and generalizations of Dedekind sums

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In 2003, we spent a lot of time in writing papers up for the results obtained in 2002.(^*) Let K be a local field and L a totally ramified Galois extension of K with [L : K] a power of p, where p is the characteristic of the residual field of K.1. Let L/K satisfy (^*) and have only one proper higher ramification group, or let L/K be of type (m, m,...,m), i. e., if G⊃_≠H1⊃_≠【triple bond】⊃_≠H_<n-1>⊃_≠{1} is a series of all the higher ramification groups of the Galois group G for L/K, then (G : H_1) =【triple bond】=(H_<n-2> : H_<n-1>)=|H_<n-1>|=m. Then we obtained towers of fields【numerical formura】and found that K = ∪^∞_<n=1>K_n has some universal property in 2002. We wrote a paper on them and submitted to a journal.2. Let L/K satisfy (*) and have exactly two proper higher ramification groups G⊃_≠H_1⊃_≠H_2⊃_≠{1} D~ H2 D~ {1} with (G : H_1)=m_1, (H_1 : H_2)=m_2 and |H_2|=m3. There are 13 cases according to sizes of m_1,m_2 and m_3, and in 2002 we obtained data for SDCs of α=π_1+π_1π_2+π_1π_2π over K in all cases, where π_1 and π_2 are prime elements of the corresponding fields to H_1 and H_2, and π is that for L.. In 2003, we wrote (and are still writing) a paper on them.3. Also we began working on the computation of H^1(K, m^^-) when char(K) = 0, where m^^-is the maximal ideal of the ring of integers of Q^^-_p. We know by Coates-Greenberg that H^1(K, m^^-)≠0, since K was shown to have a finite conductor.