Mathematical Sciences: L-Functions Associated to Siegel Modular 3-Folds

在这项研究中，Nygaard将研究2次和3权的Siegel模形式的Galois表示，最初是在Level 4的特殊情况下，但希望稍后在更一般的情况下。动机出现在Siegel模变种的上同调中，但在一些简单得多的3-折叠的上同调中暴露出来。这三重曲线是利用Riemann的theta公式构造的，它们是二次曲线的简单完全交。由于这些动机的存在，我们发现安德里亚诺夫L在西格尔模的三重哈斯-魏泽塔函数中起着因子的作用。模形式是对数论信息进行编码的解析函数，允许研究者使用深层的分析工具来求解数论函数。在这项研究中，代数几何中附着于对象的模形式进一步丰富了这一技术。私家侦探提出了一些非常令人兴奋的问题，他的努力肯定会成功。

In this research Nygaard will study Galois representations of Siegel modular forms of degree 2 and weights 3, at first in the special case of level 4 but hopefully later on in more generality. The motives occur in the cohomology of Siegel modular varieties but are exposed in the cohomology of some much simpler 3-folds. These 3-folds are constructed using the theta formulas of Riemann and are simple complete intersections of quadrics. As a consequence of the existence of these motives we find Andrianov L-functions as factors in the Hasse-Weil Zeta function of the Siegel modular 3-fold. Modular forms are analytic functions which encode number theoretic information allowing the researcher to use the deep tools of analysis to solve number theoretic functions. In this research modular forms attached to objects in algebraic geometry further enrich this technique. The P.I. poses some very exciting problems and will surely be successful in his endeavors.