Path integrals and p$p$ ‐adic L$L$ ‐functions

We prove an arithmetic ‘path integral’ formula for the inverse p$p$ ‐adic absolute values of Kubota–Leopoldt p$p$ ‐adic L$L$ ‐functions at roots of unity.


1
PRIMES, KNOTS AND QUANTUM FIELDS

Arithmetic topology
Barry Mazur [17,18] pointed out long ago that the cohomological properties of Spec(  ), the spectrum of the ring of integers of an algebraic number field, are like those of a 3-manifold.This went with the observation that the inclusion Spec(()) ↪ Spec(  ) of the spectrum of the residue field () of a prime  of   compares well to the inclusion of a knot  into a 3-manifold [22].When we remove a finite collection  of primes and consider Spec(  ) ⧵ , the properties then are like a 3-manifold with boundary obtained by removing tubular neighbourhoods of the knots.Mazur went on to consider the cases of Spec(ℤ), Spec(ℤ[1∕]), and Spec .
There, one arrives at an analogy between the covering with Galois group Γ ∶= Gal(ℚ(  ∞ )∕ℚ(  )) ≃ ℤ  and the maximal abelian covering   →  3 ⧵ , which has group of deck transformations isomorphic to ℤ.In Iwasawa theory, the main object of study is the Galois group of the maximal abelian unramified -extension  of ℚ(  ∞ ), acted on by the Iwasawa algebra The isomorphism here comes from  − 1 ↦  for a fixed topological generator  of Γ.There is also an action of Gal(ℚ(  )∕ℚ) (which can be realised as a subgroup of Gal(ℚ(  ∞ )∕ℚ)), according to which  splits into isotypic components   via the Teichmüller character  ∶ Gal(ℚ(  )∕ℚ) → ℤ ×  and its powers   .The main conjecture of Iwasawa theory [19] as proved by Mazur and Wiles relates the determinants of these isotypic components to various branches of the -adic zeta function.(For general background, we refer the reader to Washington's excellent book [27].)Namely, for an odd prime  and for  = 1, 3, … ,  − 2 odd, the Kubota-Leopoldt -adic -function   (  , ) is the continuous function on ℤ  such that   (  , ) = (1 −  − )() for negative integers  ≡  mod  − 1.There is a unique power series   () ∈ ℤ  [[𝑇]] such that   ((1 + )  − 1) =   (  , ), enabling us to identify   () with the -adic -function itself.The main conjecture says that  1− () is the determinant of the ℤ  [[]]-module   for  ≠ 1 odd.Mazur's main observation in [17] was that the Alexander polynomial of a knot also has a precise definition as a determinant of the module strengthening the circle of analogies that has now come to be known as arithmetic topology.

Quantum field theory and knots invariants
Meanwhile, in the late 1980s, Edward Witten [28] gave a remarkable construction of the Jones polynomial of a knot using the methods of quantum field theory, which were then made rigorous by Reshetikhin and Turaev [24].Here, we have a space  of (2) connections on  3 acted upon by a group  of gauge transformations.The knot  defines a Wilson loop function the trace of the holonomy of the connection around  evaluated in the standard representation  of  (2).(The importance of such a function should not be surprising at all to number-theorists.)There is also a 'global' Chern-Simons function given by which is only gauge-invariant up to integers.Witten's result is then )) , equating a path integral with the value of the Jones polynomial   of  at a root of unity.The reader should beware that this is not a theorem by any means, as the integral is not defined in a mathematical sense.Nonetheless, the physicists have fairly well-defined rules for working with such expressions, according to which Witten derives this.Furthermore, a number of different normalisations exist in the literature, about which we will not comment.It might be worth explaining a bit the terminology 'path integral'.It originates in an integral over a space of paths  ∶  →  from an interval to some space  of fields.It is often appropriate to express this as an integral over 'all possible paths' on a spacetime manifold.For example, if  represents space, then a connection on the spacetime  ×  can be thought of as a path in the space of connections on .Thus, even in the case of a more complicated spacetime and even when the manifold does not have a direction interpretation as spacetime (such as a Riemannian manifold), it has become common for physicists to refer to an integral over a suitable space of fields as being a 'path integral'.We will use this kind of terminology ourselves rather loosely in this paper in the belief that the intuition of physicists will be useful in the long run.

Arithmetic path integrals
We would like to prove a simple arithmetic analogue of Witten's formula for -adic -functions.However, it should be noted that a naive analogue of Witten's formula for the Alexander polynomial is not available in the physics/topology literature.That is, it does not suffice to take a simple path integral for abelian connections.However, somewhat elaborate integrals that give the Alexander polynomial can be found in [2,5,8,9,20,26].These are of course all inspired by Witten's formula.By comparison, our formula is indeed somewhat naive, and suggests that the topological Alexander polynomial might also have such a simple expression, even though we will not pursue this question at the moment.Following the framework of arithmetic topological quantum field theory set up in [6,7,13,15,16,23], our goal is to represent the -adic -function as an arithmetic path integral, thereby incorporating the perspective of topological quantum field theory into arithmetic topology in a rather concrete fashion and strengthening the analogy envisioned by Mazur.It should be admitted right away that we do not achieve this goal.However, we do find a result about its -adic valuation that appears to be interesting.To describe this, we go on to define the relevant space of 'arithmetic fields'.
Let  =   where  is an odd prime and  is a positive integer.We set  = ℚ(  ) and let where   is a primitive th root of unity.We fix an integer  ⩾ 1 and define the space of fields as where  1  denotes compactly supported étale cohomology [21, chapter 2].This is can be viewed as an abelian moduli space of principal bundles together with its dual, a setting that allows the definition of topological actions in physics in arbitrary dimension and even on non-orientable manifolds [8].The BF-action is the map where is the Bockstein map coming from the exact sequence is the invariant map [18].
There is a natural action of  = Gal(∕ℚ) on the space of fields ℱ  , and we let  ′ (≃ Gal(ℚ(  )∕ℚ)) ⊂  be the unique subgroup of  of order  − 1.As  − 1 is not divisible by ,  ′ acts semi-simply on ℱ  .Let us define that is, on the first factor we take the   -eigenspace, and on the second factor we take the  −eigenspace.
Theorem 1.1.Let  be odd and different from 1. Then we have Remark 1.2.Note that the variable  here corresponds to  − 1, where  is a topological generator of Γ.Thus, the value of  on exp(2∕  ) − 1 corresponds to the value exp(2∕  ) assigned to .In the Alexander polynomial, the variable  indeed corresponds to a generator of  1 ( 3 ⧵ , ℤ).Thus, in the analogy of arithmetic topology, the values occurring on the left of our formula correspond exactly to the values of the standard variable in topology at roots of unity.We could use the Weierstrass preparation theorem and the vanishing of the -invariant [10] to replace   by a distinguished polynomial   , such that for a unit   (𝑇).With this we can restore the variable  =  + 1 and put   () =   ( − 1).The left-hand side of our formula can then be written One might argue that the   are the true analogues of the Alexander polynomial.
Remark 1.3.Of course as we are taking the -adic absolute value, the formula is the same if we change   by a unit.Hence, it really is about characteristic power series rather than the precise choice of -adic -functions.Nevertheless, as the -adic -functions are the objects of central interest in number theory, we have stated the theorem in these terms.Obviously, for this we need the main conjecture of Iwasawa theory which we will take for granted in the rest of this paper.
Remark 1.4.In physics, it is common to be vague about the domain of the path integral.That is, one imagines a sequence of inclusions containing the space  of classical fields (solutions to the equation of motion) and integrals over   giving successively more information.On the other hand, after some point, further enlargement should not matter.That is, the inclusion  ⊂   into a sufficiently flabby space should play a role similar to an acyclic resolution of a complex where any two resolutions are suitably homotopic.In gauge theory, for example, the space of  ∞ connections is thought to be an adequate domain.Even there, one could include discontinuous or distributional connections in the flavour of Feynman's original heuristic arguments with jagged paths.The limit we are taking can be interpreted as an integral over the domain This has very much the flavour of a space of distributional fields.
Remark 1.5.The obvious challenge is to remove the absolute value from the -adic -value, incorporating the unit information.In the analogy with physics, a unit is like a 'phase', leading one to believe that it can be recovered from a refinement of the given path integral.For example, one might add higher order terms like  ∪  2 or even more exotic terms depending on 'coefficient cohomology classes' to the quadratic function .In a more adventurous spirit, an integral over cohomology classes, which themselves seem to resemble classical fields, might be replaced by integrals over cocycles, cochains, or derived moduli spaces of torsors.

PATH INTEGRALS FOR CYCLOTOMIC INTEGERS
Let Cl  be the ideal class group of  and  ×  the group of units in ℤ[  ][1∕(  − 1)].A repetition of the computations found in [4] shows that the arithmetic path integral Our goal will be to prove an equivariant version of this formula.
With the definitions of the previous section, we have that We will generally denote by a subscript (⋅)  the   -isotypic component of a ℤ  -module with  ′ -action.We now analyse how the action of  ′ on ℱ  interacts with the BF-functional.More precisely, we will see that the BF-functional splits: To see this, start by letting Div  be the free abelian group generated by the closed points in .Then, as is explained in [4, section 2], we have that where and As is shown in [1, section 4], the map  ∶  Using the description of the map  above (in particular, that it is equivariant), together with the non-degeneracy of the Artin-Verdier pairing, we find that Proposition 2.2.
The first equality follows from the fact that  ′ is finite étale, the second follows from the fact that the restriction of (ℳ 0 ) to  ′ is isomorphic to  ′ *    , and the last follows from the fact that taking  ′ -fixed points is an exact functor.This shows that   (, (ℳ 0 )) naturally identifies with the  0 -eigenspace of   (,    ), and we proceed by analysing the other eigenspaces.Let us note that where   is the sheaf which, under the equivalence between sheaves split by  ′′ and  ′ -modules, corresponds to ℤ∕  ℤ, but where the action of  ′ is through  − .We now claim that where   (, (ℳ 0 ))( − ) is just   ( ′ , (ℳ 0 )) as an abelian group, but with the  ′ -action twisted by  − .Indeed, the pullback of (ℳ 0 ) ⊗   to  ′ is isomorphic, as an abelian sheaf, to (ℳ 0 ), and looking at the Cech complex one finds that   ( ′ , (ℳ 0 ))( − ) is indeed isomorphic to   ( ′ , (ℳ  )).
The realisation of the path integral (2.3) as a path integral on  is now straightforward.Indeed, there is a natural Bockstein map  ∶  1 (, (ℳ  )) →  2 (, (ℳ  )) and we define as (, ) = inv( ∪ ).This realises the path integral as a path integral on , which is what we wanted to see.

CALCULATION FOR LARGE VALUES OF 𝒎
We analyse the right-hand side of the formula in Proposition 2.2 for large values of .
, then the first factor is one and the third factor equals the size of the   -isotypic component of the -primary part of Cl  .Below, we look into the factor in the middle.We will assume for simplicity that Recall that   denotes a primitive th root of unity, where  =   .Let  ′ ⊂  ×  be the subgroup generated by −  and elements of the form 1 −    where  = 1, 2, 3, … ,  − 1.Therefore, we have a finite quotient  ∶=  ×  ∕ ′ .From the exact sequence 0 →  ′ →  ×  →  → 0 and the snake lemma, we get the long exact sequence for  ⩾ , this gives an exact sequence and the same after taking   -isotypic components: Note that [  ]  ≃   [  ] and (∕   )  ≃   ∕    as | ′ | has order prime to .As  and hence   is finite, the kernel and cokernel have the same order, so that Put  ∶=  ′ ∕ 2 .As  is torsion-free, another easy snake lemma argument gives an exact sequence 0 →   →  ′ ∕( ′ )   → ∕   → 0 for  ⩾ .Here, we identified   with the cokernel of the   -power map on  2 .Thus, for  ≠ 1, we get an isomorphism ( ′ ∕( ′ )   )  ≃ (∕   )  .
Lemma 3.1.We have Proof.The structure of  as a Galois module is known.Let  + be the maximal totally real subfield of  and let  + = Gal( + ∕ℚ).In [3, Theorem 3], Bass proved that there is an isomorphism  ≃ ℤ[ + ] as Galois modules.Thus, the assertion follows because ℤ∕  ℤ[ + ] = ⨁ We interpret the right-hand side of the formula in terms of special values of the Kubota-Leopoldt -adic -functions.As in the introduction, let  = Gal(∕ℚ(  ∞ )), the Galois group of the maximal abelian unramified -extension  of ℚ(  ∞ ).We now vary  in  =   , and denote by Cl   [ ∞ ] the -primary part of the ideal class group of   ∶= ℚ(  +1 ).We assume in the following that  ≠ 1 is odd.By the main conjecture and the vanishing of the  The authors are grateful to a referee for a careful reading and many helpful suggestions.

J O U R N A L I N F O R M AT I O N
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