We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.
Deep Dive into On the p-adic Beilinson conjecture for number fields.
We formulate a conjectural p-adic analogue of Borel’s theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.
The Beilinson conjectures about special values of L-functions [2] are a far reaching generalization of the class number formula for the Dedekind zeta function. For every algebraic variety X over the rationals it predicts the leading term of the Taylor expansions of L(H i (X), s) at certain points, up to a rational multiple, in terms of arithmetic information associated with X, namely, its algebraic K-groups K j (X) [43]. More generally, these conjectures can also be formulated for motives.
There have been several important steps taken towards verification of these conjectures in various cases, although, strictly speaking they have only been verified completely in the case where X is the spectrum of a number field, where they follow from famous theorems of Borel [10,11].
To motivate what follows, let us briefly recall the conjecture that interests us the most (for an introduction see [23,46]). One associates with X two cohomology groups. The first one is the Deligne cohomology H i D (X /R , R(n)), which is an Rvector space. The second is “integral” motivic cohomology H i M (X /Z , Q(n)), which may be defined as a certain subspace of K 2n-i (X) ⊗ Z Q. There is a regulator map defined by Beilinson, (1.1)
has a rational structure coming from the relations between H i D (X R , R(n)) and the de Rham and singular cohomology groups of X [46, p.30].
The first part of the conjecture is that the map in (1.1) induces an isomorphism between the lefthand side tensored with R and the righthand side, and consequently provides a second rational structure on det H i D (X /R , R(n)). The second part of the conjecture states that, assuming a suitable functional equation for L(H i-1 (X), s), these two rational structures differ from each other by the leading term in the Taylor expansion of L(H i-1 (X), s) at s = in. Because of the expected functional equation one can reformulate the conjecture in terms of L(H i-1 (X), n) (see [2,Corollary 3.6.2] or [31, 4.12]).
As mentioned before, this conjecture has only been verified in the case of number fields, due to difficulties in the computation of motivic cohomology. What has been verified in several other cases is a form of the conjecture in which one assumes the first part. For this one finds dim R H i D (X /R , R(n)) elements of H i M (X /Z , Q(n)), checks that their images under (1.1) are independent, hence should form a basis of H i M (X /Z , Q(n)) according to the first part of the conjecture, and verifies the second part using these elements.
The idea that there should be a p-adic analogue of Beilinson’s conjectures has been around since the late 80’s. Such a conjecture was formulated and proved by Gros in [29,30] in the case of Artin motives associated with Dirichlet characters. In the weak sense mentioned before, it was proved for certain CM elliptic curves in [17] (where the relation with the syntomic regulator is proved in [4] and further elucidated in [7]), and for elliptic modular forms it follows from Kato’s work (see [47]).
The book [42] contains a very general conjecture about the existence and properties of p-adic L-functions, from which one can derive a p-adic Beilinson conjecture. Rather than explain this in full detail we shall give a sketch of this conjecture similar to the sketch above of the Beilinson conjecture.
For the p-adic Beilinson conjectures one has to replace Deligne cohomology with syntomic cohomology [30,41,3], the Beilinson regulator with the syntomic regulator, and L-functions with p-adic L-functions.
Syntomic cohomology H i syn (Y, n) is defined for a smooth scheme Y of finite type over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. For the Q-variety X we obtain, for all but finitely many primes, a map (1.2)
where Y is a smooth model for X over Z p . This cohomology group is a Q p -vector space.
The theory of p-adic L-functions starts with Kubota and Leopoldt’s p-adic ζfunction [35], ζ p (s), which is defined by interpolating special values of complex valued Dirichlet L-functions. This principle has been extended to ζ-and L-functions in various situations, resulting in corresponding p-adic functions for totally real number fields [1,15,22], CM fields [32,33] and modular forms [37]. (Given the occasion, let us note that the approach of Deligne and Ribet using modular forms was initiated by Serre [48].)
The p-adic Beilinson conjecture therefore has many similarities with its complex counterpart. However, there is a very important difference. In general, when 2n > i+1, there is no hope that (1.2) induces an isomorphism after tensoring the lefthand side with Q p . To see this, consider a number field k with ring of algebraic integers O k . By Borel’s theorem (see Theorem 2.2) we have, in accordance with Beilinson’s conjectures,
when n ≥ 2 is even r 1 + r 2 when n ≥ 2 is odd with r 1 (resp. 2r 2 ) the number of real (resp. complex) embeddings of k. Thus, motivic cohomology “knows” about the num
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