A note on abelian arithmetic BF-theory

We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $\mathbb{G}_m$ and abelian varieties.


Proposition 1.1.
(a,b)∈H 1 (X,Z/n)×H 1 (X,µn) Thus, for n large enough, the 'finite path integral' will capture exactly |O × X /(O × X ) n | · | Cl F |, a quantity of the form (regulator × class number). The precise notation and definitions that go into this proposition as well as the next one will be explained in subsequent sections. Now let A and B be dual abelian varieties over F with semi-stable reduction at all places. Let n be an integer coprime to all the Tamagawa numbers of A and B as well as to the places of bad reduction for them. Assume that the Tate  We view these formulae as some (weak) evidence for the suggestion made in [9] that an arithmetic topological functional on moduli spaces of Galois representations will have something to do with Lfunctions. In any case, it is rather striking to find expressions for the orders of class groups and Tate-Shafarevich groups as exponential sums, which perhaps have not appeared heretofore in the literature. Equally notable is that a path integral for the BF functional for three manifolds leads to the Alexander polynomial of a knot in the physics literature [5], which is well-known to be an analogue of the p-adic L-function [15].
In forthcoming work, we will develop BF -theory for arithmetic schemes with boundary, that is Spec(O F [1/S]) for a finite set S of places, as well as a p-adic theory. Also interesting would be to develop arithmetic BF -theory for arithmetic schemes in higher dimension by way of the duality theory of [8]. However, in the present announcement, the primary goal is to illustrate with a minimum of clutter the relationship between a path integral in the sense of physicists and important arithmetic invariants.
2. Some finite path integrals for G m : Proof of Proposition 1.1.
As before, let X = Spec(O F ) for F a totally imaginary number field and let µ n and Z/n be the usual finite flat group schemes viewed as sheaves in the flat topology. We have the Bockstein map coming from the exact sequence 1 ✲ µ n ✲ µ n 2 ✲ µ n ✲ 1 and the invariant isomorphism [12] : Define the BF-functional on (The class δb is the analogue of the curvature F .) Remark 2.1. One can also define a BF-functional on M as BF ′ (a, b) = (δa∪b), where δ : H 1 (X, Z/n) → H 2 (X, Z/n 2 ) is the Bockstein map coming from the exact sequence However, BF ′ = BF, since we have an equality δa ∪ b = a ∪ δb by Lemma 2.1 and the proof of Lemma 2.2 in [7].
We will now calculate the path integral First, let us calculate the groups H i (X, µ n ). We define Div F to be the group of fractional ideals of F and for x ∈ F * , we let div(x) be the associated principal ideal. We claim that To see this, note that µ n , is quasi-isomorphic to the complex with non-zero terms in degree 0 and 1. Since G m,X is representable by a smooth group scheme, flat cohomology groups with coefficients in G m,X coincide with the corresponding étale cohomology groups [14, III, Theorem 3.9]. Thus, to compute flat cohomology with coefficients in µ n , it suffices to compute the hypercohomology of C • , viewed as a complex of sheaves on the étale site. To do this, we take the well-known resolution is the inclusion of the generic point and where Z /x is the skyscraper sheaf at the closed point x. Resolving the complex C • levelwise with this resolution and taking the total complex of the resulting double complex, we get the complex

Now, as in Lemma 4.2 and Corollary 4.3 of [4]
one uses the local-to-global spectral sequence to conclude that the complex above computes the flat cohomology of µ n for i = 0, 1, 2; to conclude the calculation for i ≥ 3 one applies flat duality as in III, Corollary 3.2 of [13]. We now proceed to analyze the Bockstein map δ : H 1 (X, µ n ) ✲ H 2 (X, µ n ).
Using the complex given earlier, the Bockstein was computed in Lemma 4.1 of [1], and it is the composite of two maps: the first is the surjective map which takes (x, I) ∈ H 1 (X, µ n ) ∼ = Z 1 /B 1 to I ∈ Cl F [n] and the second is the reduction map Cl F [n] → Cl F /n. By noting that the kernel of the first map is O × X /(O × X ) n and that the kernel of the second map is n Cl F [n 2 ], we see that our sum becomes exp(2πia ∪b) whereb ∈ Cl F /n is the reduction of b. But for b non-trivial, it is clear that this sum is zero, giving us In particular, if Cl F [n] = Cl F , we see that the sum evaluates to |O × X /(O × X ) n | · | Cl F |. If one calculates the path integral (a,b)∈M exp (2πiBF ′ (a, b)) where BF ′ is as in the above remark, one sees, since BF ′ coincides with BF, that (a,b)∈M exp (2πiBF (a, b)) equals | Cl F [n]| · |O × X /(O × X ) n | times the number of isomorphism classes of unramified Z/n-étale algebras which can be embedded into an Z/n 2 -étale algebra. where the left hand side is flat cohomology (in the fppf site). We claim that our assumption that B has semi-stable reduction, together with our assumptions on n, implies that multiplication by n on B is an epimorphism in the category of fppf sheaves. To see this, note that by [6,Lemma B.4.], B 0 ·n − → B 0 is faithfully flat, thus an epimorphism of fppf sheaves. Further, we have a commutative diagram and since n is prime to |Φ B |, multiplication by n on Φ B is actually an isomorphism. Thus, by the snake lemma, multiplication by n on B is an epimorphism. By [13,Corollary 3.4] there is a perfect pairing We now define the BF-functional where is the isomorphism H 3 (X, µ n ) ∼ = 1 n Z/Z, as in the previous section. Our goal is now to calculate the path integral  2πiBF (a, b)).
To this end, note that, by [ Assuming the finiteness of the Tate-Shafarevich group, we can take n large enough so that By an argument identical to the case of µ n above, since