Fermat Adeles and Differential Modular Forms

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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.