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

2003年，我们花了大量时间为2002年的结果撰写论文。(^*)设K是局部域，L是K的完全分枝伽罗瓦扩张，[L：K]是p的幂，其中p是K1的剩余域的特征。设L/K满足(^*)且只有一个真高分支群，或设L/K为(m，m，…，m)型，即：若G⊃_≠H1⊃_≠[三重键]⊃_≠H_&lt；n-1&gt；⊃_≠{1}是L/K的Galois群G的所有高分支群的系列，则(G：H_1)=[三重键]=(H_&lt；n-2&gt；：H_&lt；N-1&gt；)=|H_&lt；n-1&gt；|=m，得到了场塔[数值形式]，发现2002年K=∪^∞_&lt；n=1&gt；K_n具有一定的普适性。我们写了一篇关于他们的论文，并提交给了一家报刊。设L/K满足(*)且恰好有两个真的高阶分支群G⊃_≠H_1⊃_≠H_2⊃_≠{1}D~H2 D~{1}且(G：H_1)=m_1，(H_1：H_2)=m_2和|H_2|=m_3.根据m_1、m_2和m_3的大小有13种情况，2002年我们得到了所有情况下α=π_1+π_1π_2+π_1π_2π的数据，其中π_1和π_2是H_1和H_2对应的域的素元，π是L的素元。2003年，我们写了一篇关于它们的论文(现在仍在写)。当≠(K)=0时，我们开始计算H^1(K，m^-)，其中m^-是q^-p的整数环的极大理想。我们从Coates-Greenberg那里知道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.