Study of modulimap of degenerate families of algebraic curves and local signature arizing from automorphic form

2005年，我们在退化代数曲线族的局部特征和Horikawa指数方面取得了以下进展：（1）稳定族的情况;我们已经在2004年发现了Harris-Mamford公式在代数曲线族的最大角纤维化中的应用。在2005年，我们发现了Eisenbud-Harris公式在甚至亏格情况下的一个类似应用。在亏格为4的情况下，我们从Cheu-Konno下界得到了一个局部贡献公式，但在亏格大于等于6的情况下，由于需要Eisenbud-Harris公式的更多Sharp形式，所以还存在一些困难.（2）不稳定族的情形：我们在2004年已经用单值数据的Dedekind公式描述了不变量在稳定约化下的行为，2005年我们得到了三个Dedekind和项的完全显式公式，因此我们可以把以前的公式改写成一个简单的形式，从而使关于亏格的归纳成为可能。从数论的观点来看，这似乎很有趣。

In 2005, we have the following development about the local signature and Horikawa index of degenerate families of algebraic curves :(1)The case of stable family ; We have already found in 2004 an application of Harris-Mamford formula to maximal-gonal fibration of addgeues. In 2005, We have found a similar application of Eisenbud-Harris formula to even genus case. More precisely, in the case of genus 4, we obtain a formula for local contribution from the Cheu-Konno lower bound But in the case of genus greater than or equal 6, some difficulties remain, because we need more Sharp form of Eisenbud-Harris formula.(2)The case of unstable family ; We have already describe in 2004 the behavionr of invariants under the stable reduction by using the Dedekind sam of monodromy data In 2005, we have obtained completely explicite formula for there Dedeking sum tern Therefore we can rewrite out previous formula in a simple form so that the induction with respect to the genus in possible. It seems to be interesting from the number theoretic viewpoint.