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

在2003年，我们花了很多时间在写论文，为2002年取得的成果。（^*）设K是局部域，L是K的全分歧伽罗瓦扩张，[L：K]是p的幂，其中p是K的剩余域的特征。设L/K满足（^*）且仅有一个真高阶分歧群，或L/K为（m，m，. m）、i.例如，若G_H {<n-1>1}是Galois群G对L/K的所有高阶分支群的级数，则（G：H_1）=[三键]=（H_<n-2>：H_<n-1>）=| H_<n-1>|=m。然后在2002年我们得到了场塔[数值公式]，并发现K = K ^∞_&lt;n=1&gt;K_n具有一些普适性质。我们写了一篇关于它们的论文，并提交给一家杂志。令L/K满足（*）且恰好有两个真高级分歧群G_≠H_1_≠H_2_≠{1} D~ H2 D~ {1}，其中（G：H_1）=m_1，（H_1：H_2）=m_2且|氢气|=m3。2002年我们得到了K上α=π 1 +π 1 π 2 +π 1 π 2 π的SDCs的数据，其中π 1和π 2是H1和H2对应域的素元，π是L的素元. 2003年，我们写了（现在还在写）一篇关于它们的论文。我们也开始计算当char（K）= 0时的H^1（K，m^^-），其中m^^-是Q^^-_p的整数环的极大理想。我们通过科茨-格林伯格知道H^1（K，m^^-）= 0，因为K被证明有一个有限导体。

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.