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)，其自变量由一个素数p和一个整数元组a组成。(典型的例子是勒让德符号f(a，p)=(a/p)；更一般的例子是依赖于参数a的各种代数几何对象的L函数的p系数。)这样的函数f通常不是(p，a)中的多项式，因此它们“超越了代数几何的语言”。这个提议的第一个主要思想是通过一个新的运算--‘’伴随‘’来扩展通常的代数几何--‘’费马商运算‘’，该运算在有理数的Adeles上的作用如同将Adele的第p个入口x(P)送入关于p的x(P)的费马商。提出的主要猜想原理是，这个更大几何的函数环(称为费尔马特-阿德莱斯环)可以用来‘表示’许多如上所述的有趣的算术函数f。对于每一个固定的p，研究人员在他以前的研究中已经引入了这样的几何；本提议的主要任务是使p按几何方式变化。这种方法的策略，至少对于Abelian变种，是使用Siegel微分模形式(在这个更大的几何中，这是通常的Siegel模形式的类似)。Siegel微分模形式的研究是该提案的第二个主要主题。研究Siegel微分模形式的不同动机可以描述如下。在维g的主极化阿贝尔格式的‘’模空间‘’A上，存在‘’同源‘’给出的一个自然等价关系。商A/~在任何合理的意义上都不存在于通常的代数几何中，但它在更大的“费马几何”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，其方法类似于由通常的Siegel模形式给出的A到射影空间的嵌入。于是，A/~的“射影几何”和“上同调”就产生了一系列问题。这一观点可以推广到其他商X/~，其中X是整数上的有限类型方案，~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质数的各种量如何随着质数的变化而变化。这些量的变化不受经典代数几何的支配，因为这些量不是变量素数的多项式函数或代数函数。这位研究人员提议开发一种新的几何学来描述这种变化。这种几何将从经典的代数几何中通过加上另一个运算--费马商--而得到。一旦几何以这种方式放大，一系列在经典代数几何中没有任何几何意义的令人费解的商对象开始在几何上有意义。这种在算术代数几何中看待算术函数和商问题的特殊方式应该会给这些对象的研究带来一种新的几何直觉。

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.