Fermat Adeles and Differential Modular Forms

数论中一些最有趣的对象被编码成整数值函数f（p，a），其参数由素数p和整数元组a组成。 (The典型的例子是f（a，p）=（a/p），勒让德符号;更一般的例子是由依赖于参数a的各种代数几何对象的L函数的p系数给出的。 这样的函数f一般不是（p，a）中的多项式，因此它们“超越了代数几何的语言”。 该建议的第一个主要思想是通过"附加“一个新的运算，即”费马商运算“，来扩大通常的代数几何，通过将adele的第p个项x（p）发送到x（p）关于p的Fermat商中来起作用。提出的主要几何原理将是，这个较大几何的函数环（称为费马阿戴尔斯环）可以用来“代表”许多有趣的算术函数f如上所述。 对于每一个固定的p，这样的几何形状已经介绍了调查员在他以前的研究，在目前的建议的主要任务是使p变化“几何”。 该方法的策略，至少对于阿贝尔变种，是使用西格尔微分模形式（这是一个模拟，在这个更大的几何，通常西格尔模形式）。 西格尔微分模形式的研究是该提案的第二个主题。 一个不同的动机研究西格尔微分模形式可以描述如下。 在g维主极化阿贝尔概型的模空间A上，存在一个由同构给出的自然等价关系。在通常的代数几何中，商A/~在任何合理的意义上都不存在，但在更大的“费马三角几何”中有一个很好的替代品。 A/~到射影空间的嵌入应该由Siegel微分模形式给出，其方式类似于A到射影空间的嵌入，由通常的Siegel模形式给出。一系列的问题，然后出现的“射影几何”和"上同调“的A/~。 这一观点可以推广到其他的等价关系X/~，其中X是整数上的有限类型的方案，~是X上的“算术定义的等价关系”。数论的主要问题之一是理解自然依赖于素数的各种量如何随着素数的变化而变化。 这些量的变化不受经典代数几何的支配，因为这些量不是可变素数的多项式或代数函数。 研究人员建议开发一种新的几何形状来描述这种变化。 这种几何可以从经典代数几何中通过附加一个运算，即费马商得到。 一旦几何以这种方式被扩大，一系列在经典代数几何中没有任何几何意义的令人困惑的商对象就开始有几何意义了。 这种特殊的方式来看待算术函数和商的问题，算术代数几何应该带来一个新的，几何，直觉到研究这些对象。

Some of the most interesting objects in number theory are encoded into integer valued functions f(p,a) whose arguments consist of a prime p and a tuple of integers a. (The prototypical example is f(a,p)=(a/p), the Legendre symbol; more general examples are given by p-coefficients of L-functions of various algebraic-geometric objects depending on parameters a.) Such functions f are generally not polynomials in (p,a) hence they ``transcend the language of algebraic geometry''. The first main idea of the proposal is to enlarge usual algebraic geometry by ``adjoining'' one new operation, the ``Fermat quotient operation'' which, on the adeles of the rational numbers, acts as by sending the p-th entry x(p) of the adele into the Fermat quotient of x(p) with respect to p. The main conjectural principle proposed will be that the ring of functions of this larger geometry (called the ring of Fermat adeles) can be used to ``represent'' many of the interesting arithmetic functions f as above. For each fixed p, such a geometry has been introduced by the investigator in his previous research; the main task in the present proposal is to make p vary ``geometrically''. The strategy of the approach, at least for Abelian varieties, is to use Siegel differential modular forms (which are an analogue, in this larger geometry, of usual Siegel modular forms). The study of Siegel differential modular forms is the second main theme of the proposal. A different motivation for the study of Siegel differential modular forms can be described as follows. On the ``moduli space'' A of principally polarized Abelian schemes of dimension g there is a natural equivalence relation ~ given by ``isogeny''. The quotient A/~ does not exist, in any reasonable sense, in usual algebraic geometry, but it has a nice substitute in the larger, ``Fermat adelic geometry''. The embedding of A/~ into a projective space should be given by Siegel differential modular forms in a way similar to the embedding of A into a projective space, given by usual Siegel modular forms. A series of problems then arise as to the ``projective geometry'' and ``cohomology'' of A/~. This point of view can be generalized to other quotients X/~ where X is a scheme of finite type over the integers and ~ is an ``arithmetically defined equivalence relation'' on X.One of the main problems of number theory is to understand how various quantities that naturally depend on prime numbers vary as the prime number varies. The variation of these quantities is not governed by classical algebraic geometry in the sense that these quantities are not polynomial or algebraic functions of the variable prime number. The investigator proposes to develop a new geometry that would describe this variation. This geometry would be obtained from the classical algebraic geometry by adjoining one more operation, the Fermat quotient. Once geometry has been enlarged in this way, a series of puzzling quotient objects that did not have any geometric meaning in classical algebraic geometry start making sense geometrically. This particular way of looking at arithmetic functions and quotient problems in arithmetic algebraic geometry should bring a new, geometric, intuition into the study of these objects.