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签名和Horikawa指数方面有了以下发展：(1)稳定族的情况；2004年，我们已经发现Harris-Mamford公式在加法的极大平方纤维上的应用。2005年，我们发现Eisenbud-Harris公式在偶数亏格情形中也有类似的应用。更确切地说，对于亏格4，我们从Cheu-Konno下界得到了一个局部贡献公式，但对于亏格大于等于6的情况，仍然存在一些困难，因为我们需要更多的Sharp形式的Eisenbud-Harris公式。(2)不稳定族的情况；我们已经在2004年描述了不变量在稳定约化下的行为，利用2005年的Mondromy数据，我们得到了关于Dedeking和的完全显式公式，因此我们可以用一个简单的形式改写以前的公式，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.