Geometric main conjectures in function fields

The set 𝑆 is called 𝑝-large if the following are satisfied.

1. Pic S 0 ⁢ ( L ) p = 0 .

2. 𝑆 contains at least one place of degree (relative to F q ) coprime to 𝑝.

It is easily seen that 𝑆 is 𝑝-large if and only if Pic S ⁢ ( L ) p = 0 , where

is the quotient of the full Picard group Pic ⁢ ( L ) of 𝐿 by its subgroup of 𝑆-supported divisor classes. The equivalent formulation is perhaps a more natural definition of 𝑝-largeness, but we prefer to use the definition above because Pic S 0 ⁢ ( L ) (as opposed to Pic S ⁢ ( L ) ) is much more naturally related to the 1-motive M S .

Proposition A.3 (Greither–Popescu [7])

If 𝑆 is 𝑝-large, then the following hold.

1. There is a canonical isomorphism T p ⁢ ( M S ) Γ ≃ U S ⁢ ( L ) p .

2. There is a canonical isomorphism T p ⁢ ( M S ) Γ ≃ X S ⁢ ( L ) p .

In fact, in [7], the authors describe the modules T p ⁢ ( M S , T ) Γ , T p ⁢ ( M S , T ) Γ , where M S , T is the Picard 1-motive associated to ( Z ̄ , F q ̄ , S ̄ , T ̄ ) , where 𝑇 is a finite, non-empty set of closed points on 𝑍, disjoint from 𝑆. However, by [8, Remark 2.7] and the proof of [7, Lemma 3.2], we have for any such 𝑇 equalities

where U S , T ⁢ ( L ) is the group of 𝑆-units in L × , congruent to 1 modulo all primes in 𝑇.

With notation as above, the following hold for all sets 𝑆.

1. There is a canonical isomorphism T p ⁢ ( M S ) Γ ≃ U S ⁢ ( L ) p .

2. There are canonical exact sequences of Z p -modules,

   0 → Pic S 0 ⁢ ( L ) p → T p ⁢ ( M S ) Γ → X ̃ S ⁢ ( L ) → 0 , 0 → Z p / d S ⁢ Z p → X ̃ S ⁢ ( L ) → X S ⁢ ( L ) p → 0 ,

   where d S ⁢ Z = deg ⁢ ( Div S ⁢ ( L ) ) and

   X ̃ S ( L ) : = ( Div S ̄ 0 ( L ̄ ) p ) Γ .

   In particular, if 𝑆 contains a prime of degree not divisible by 𝑝, then d S ∈ Z p × and X ̃ S ⁢ ( L ) ≃ X S ⁢ ( L ) p .

Proof

We will give only a brief sketch of the proof, as the techniques and main ideas are borrowed from [7]. First, we consider the exact sequence of Z p ⁢ [ [ Γ ] ] -modules

and take Γ-invariants and Γ-coinvariants to obtain a long exact sequence

The fact that the Γ-invariant of the complex [ Div S ̄ ⁢ ( L ̄ ) p → deg Z p ] is [ Div S ⁢ ( L ) p → deg Z p ] and its Γ-coinvariant is

follows immediately from the definitions and is explained in [7, §2]. Now, in the long exact sequence above, we have

Therefore, if we let X ̃ S ( L ) : = ( Div S ̄ 0 ( L ̄ ) p ) Γ , we have a canonical exact sequence

If 𝐽 denotes the Jacobian of Z ̄ , there is a canonical exact sequence of Z p -modules

(See [8, §2] for the exact sequence above.) Since we have a canonical isomorphism

(see [8, Corollary 5.7]) and T p ⁢ ( J ) is Z p -free of finite rank, we also have T p ⁢ ( J ) Γ = 0 . Therefore, when taking Γ-invariants and Γ-coinvariants in the above exact sequence, we obtain a canonical long exact sequence of Z p -modules

where the connecting morphism 𝛿 is the usual divisor-class map. (See [7, §1] for this fact.) As there is a canonical isomorphism U S ⁢ ( L ) p ≃ ker ⁡ ( δ ) , where U S ⁢ ( L ) p injects into Div S 0 ⁢ ( L ) p via the divisor map, we obtain a canonical isomorphism of Z p -modules T p ⁢ ( M S ) Γ ≃ U S ⁢ ( L ) p , which concludes the proof of part (1) of the proposition.

To conclude the proof of part (2), observe that, by definition, coker ⁡ ( δ ) = Pic S 0 ⁢ ( L ) p . Therefore, the last four non-zero terms of the long exact sequence above lead to a canonical short exact sequence of Z p -modules

In combination with (A.1), this concludes the proof of part (2). ∎

Remark A.6 (Ritter–Weiss modules and Tate sequences)

Let L / K be a finite Galois extension with group 𝐺 and assume that F q ⁢ ( t ) ⊆ K ⊆ L . Further, assume that the set 𝑆 is 𝐺-equivariant. Then all the Z p -modules involved in the proof of the above proposition carry natural Z p ⁢ [ G ] -module structures. Most importantly, due to their canonical constructions, all the exact sequences above are exact in the category of Z p ⁢ [ G ] -modules.

Exact sequence (A.2) is the 𝑝-adic, function field analogue of the Ritter–Weiss exact sequence (see [19]), defining a certain extension class ∇ S of a module of 𝑆-divisors by an 𝑆-ideal class group, in the number field setting. This is what prompts the notation

Further, since T p ⁢ ( M S ) is Z p ⁢ [ G ] -projective, the exact sequence of Z p ⁢ [ G ] -modules

is the 𝑝-adic, function field analogue of a Tate exact sequence (see [7] and also [19] for more details), in the case where 𝑆 is not necessarily 𝑝-large.

Of course, in order to cement these analogies, one would have to compute the extension classes of (A.2) and (A.3) in Ext Z p ⁢ [ G ] 1 ⁡ ( X ̃ S ⁢ ( L ) , Pic S 0 ⁢ ( L ) p ) and Ext Z p ⁢ [ G ] 2 ( ∇ S ( L ) p , U S ( L ) p ) and show that they coincide with the class-field theoretically meaningful Ritter–Weiss and Tate classes, respectively. In [7], this was done ℓ-adically, for ℓ ≠ p , for the exact sequence (A.3), in the case where 𝑆 is ℓ-large. (See [7, Theorem 2.2].) A proof of the 𝑝-adic analogue of that theorem (even in the case where 𝑆 is 𝑝-large) is still missing in the literature, unless | G | is not divisible by 𝑝, in which case this was proved in [7]. (See [7, Theorem 2.2].)