論文の公開元へOn plectic and p-adic Hodge theory (本文)
概要
0. OVERVIEW
Main targets of arithmetic geometry are algebraic varieties, spaces defined by polynomial equa- tions, and relations between invariants of algebraic varieties and the number theory. Cohomology is a powerful method of extracting the information of algebraic varieties in terms of linear alge- bra; that is, for an algebraic variety X, one can often associate a family of vector spaces of the form Hn(X) n∈N. Today, many kinds of cohomology theories, i.e. constructions of such fam- ilies, have established depending on the situation. While all cohomology theories have similar behavior (e.g. dimension of Hn(X)), they often have different additional structures.
For example, for a complex algebraic variety X, one can associate the Betti (singular) coho-mology Hn(X, R) and the C ∞-de Rham cohomology Hn (X, R) for n ∈ N as finite dimensional R-vector spaces. The additional structures on these cohomology groups for X which is projec- tive are studied by Hodge, de Rham, Dolbeault, etc. as the classical Hodge theory in 1930’s, and Deligne generalized it to any X. As a central result, it is known that these cohomology groups together define a mixed Hodge structure (Definition 1.4) through a canonical isomorphism Hn(X, R) ∼= Hn (X, R).
As an application of the Hodge theory to the number theory, Deligne and then Beilinson formulated conjectures that describe the special values of L-functions, which are important subjects of the number theory. The Beilinsion conjecture states that, for any smooth projective algebraic variety X over Q and integers k and i with k > i + 1, the value of the L-function L(hi(X), s) at s = k is equal to a certain Hodge theoretic invariant (the determinant of an R-linear map), up to multiplication by non-zero rational numbers. For example, the L-function of the point Spec Q is the Riemann zeta function ζ(s) = L(h0(Spec Q), s) = n≥1 n−s, and it is known that the values of ζ(s) at positive even integers are of the form ζ(n) = aπn with a Q×. The values of ζ(s) at positive odd integers may not be written as simply as those at positive even integers, but the conjecture has also formulated and proved in this case. Today, the Beilinson conjecture is solved only for a few cases, namely for points, some curves, and some surfaces. It is known that a method of the polylogarithms is very useful, however, it does not work in higher dimensional cases. This problem arises from the fact that the higher extension groups of mixed Hodge structures are trivial.
In order to give a new approach to the special values of L-functions, Nekov´aˇr and Scholl recently proposed a new theory concerning further additional structures on cohomology. Their theory is innovative and beautiful, but still largely conjectural. They speculated that, if an algebraic variety (or more precisely a motive) admits real multiplication by a totally real field, then cohomology groups have a further additional structure, called plectic structure. The advantage of considering plectic structure is that nontrivial higher extensions would arise. Therefore one may expect to obtain refined invariants by using the polylogarithms, and apply it to the study of the special values of L-functions.
Let us consider Nekov´aˇr and Scholl’s theory for complex algebraic varieties. We first note that there exists an affine group scheme and an equivalence MHSR ∼= RepR( ) between the category of mixed R-Hodge structures and the category of finite dimensional R-linear representations of. In light of this, they conjectured that the cohomology groups of a complex algebraic variety with real multiplication by a totally real field would carry an R-linear action of g, where g is the degree of the totally real field and g is the g-ple selfproduct of . However, there is still a question of how one can give such a structure on the cohomology group in practice. Considering certain algebraic tori and abelian varieties as examples, one can observe (as in Section 2) that it would be possible to give an increasing filtration W• and descending filtrations F •, . . . , F • on the cohomology group. Therefore we now need to define a category MHSg in terms of filtrations W , F •, . . . , F • with an equivalence MHSg ~ Rep problem, and moreover confirm that the category RepR( g) admits nontrivial higher extension, by constructing explicit complexes calculating the extension groups. The results in Part 1 are given by a joint work with Kenichi Bannai, Kei Hagihara, Shinichi Kobayashi, Shuji Yamamoto, and Seidai Yasuda in [2]. This is the first step of the germinating study of the plectic Hodge theory.
As a development in another direction, the p-adic Hodge theory, the study of cohomology theories for algebraic varieties over p-adic fields, was started by Tate, Grothendieck, etc. from 1960’s, and developed by Fontaine, Faltings, Tsuji, etc. Let p be a prime number and K a finite extension of Qp. Then the valuation ring V of K has a unique prime ideal m and k := V/m is a finite field of characteristic p. For a smooth algebraic variety X over K, we may obtain an algebraic variety Y over k by considering modulo m of the equations defining X, which we call a reduction of X. We note that a reduction Y can have singularity even if X is smooth, and the behavior of cohomology groups of X varies by the singularity of the reduction. The most essential situation is that a reduction Y has at most normal crossing singularity. In this case, on the one hand, one can consider the log crystalline cohomology Hn (Y/W 0) over a certain base log scheme, which is endowed with two endomorphisms as additional structures. On the other hand, the algebraic de Rham cohomology Hn (X) admits a natural filtration as an additional dR crys n structure. Furthermore, Hyodo and Kato constructed a map Ψπ : Hcrys(Y/W ) → HdR(X) depending on a choice of a prime element π V , which induces an isomorphism after suitable base change. Identifying those cohomology groups via Ψcrys, we obtain an object called an admissible filtered (φ, N )-module (Definition 11.2), as a p-adic analogue of a mixed Hodge structure.
As well as the Hodge theory plays an important role in the study of (the transcendental part of) the special values of L-functions, the p-adic Hodge theory is applied to the study of the special values of p-adic L-functions, which relate with the rational part of the special values of L-functions. However, the construction of the map Ψcrys is highly technical and not p-adically analytic. Owing to this, it would be difficult to relate the p-adic L-functions with cohomological invariants. For this reason, we now cast a spotlight on another map ΨGK : Hn (Y/W 0) →n (X) constructed by Große-Kl¨onne, where the log rigid cohomology is used instead of the log crystalline cohomology. Since the log rigid cohomology is p-adically analytic, it would be useful for future application to p-adic polylogarithms and p-adic L-functions. However, although the construction of ΨGK is natural, its functoriality is nontrivial and left unproven due to technical π crys GK reasons. In addition, the comparison between Ψπ and Ψπ is also left. In this thesis, we will solve these problems by improving cohomology theory for log schemes with boundary. The results in Part 2 are based on a joint work with Veronika Ertl in [25]. However, some technical arguments are simplified by using a generalization of weak formal schemes in [26].
We note that Part 1 and Part 2 are logically independent of each other. Their combination, namely the plectic p-adic Hodge theory, should be studied, but we do not deal with it in this thesis. We will first review the classical Hodge theory in Section 1. Then we will explain more precise background of the plectic Hodge theory in Section 2 and the p-adic Hodge theory in Section 3, where we will also review the contents of Part 1 and Part 2, respectively.
