ASPECTS OF IWASAWA THEORY OVER FUNCTION FIELDS 1 ...

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IWASAWA THEORY OVER FUNCTION FIELDS 10Lemma 3.3. Let F d ⊂ F e be an inclusion of multiple Z p -extensions, e > d. Assume thatA[p ∞ ](F) = 0 or that F itt Λ(Fd )(S(F d )) is principal. Thenπ FeF d(F itt Λ(Fe)(S(F e ))) ⊆ F itt Λ(Fd )(S(F d )) .Proof. Consider the natural map a e d : Sel A(F d ) p → Sel A (F e ) Γe dpand dualize to getwhere (as in Theorem 2.6)S(F e )/I e d S(F e) → S(F d ) ↠ (Ker a e d )∨Ker a e d ↩→ H1 (Γ e d , A[p∞ ](F e ))is finite. If A[p ∞ ](F) = 0 then (Ker a e d )∨ = 0 andIf (Ker a e d )∨ ≠ 0 one hasπ e d (F itt Λ e(S(F e ))) = F itt Λd (S(F e )/I e d S(F e)) ⊆ F itt Λd (S(F d )) .F itt Λd (S(F e )/I e d S(F e))F itt Λd ((Ker a e d )∨ ) ⊆ F itt Λd (S(F d )) .The Fitting ideal of a finitely generated torsion module contains a power of its annihilator,so let σ 1 , σ 2 be two relatively prime elements of F itt Λd ((Ker a e d )∨ ) and θ d a generator ofF itt Λd (S(F d )). Then θ d divides σ 1 α and σ 2 α for any α ∈ F itt Λd (S(F e )/I e d S(F e)) (it holds,in the obvious sense, even for θ d = 0). Henceπ e d (F itt Λ e(S(F e ))) = F itt Λd (S(F e )/I e d S(F e)) ⊆ F itt Λd (S(F d )) .Remark 3.4. In the case A = E an elliptic curve, the hypothesis E[p ∞ ](F) = 0 is satisfiedif j(E) ∉ (F ∗ ) p , i.e., when the curve is admissible (in the sense of [5]); otherwise j(E) ∈(F ∗ ) pn − (F ∗ ) pn+1 and one can work over the field F pn . The other hypothesis is satisfied ingeneral by elementary Λ(F d )-modules or by modules having a presentation with the samenumber of generators and relations.Let π Fd be the canonical projection from Λ to Λ(F d ) with kernel I Fd . Then the previouslemma shows that, as F d varies, the (π Fd ) −1 (F itt Λ(Fd )(S(F d ))) form an inverse system ofideals in Λ.Definition 3.5. Assume that A[p ∞ ](F) = 0 or that F itt Λ(Fd )(S(F d )) is principal for any F d .Define˜F itt Λ (S(F)) := lim←−(π Fd ) −1 (F itt Λ(Fd )(S(F d )))Fdto be the pro-Fitting ideal of S(F) (the Pontrjagin dual of Sel E (F) p ).Proposition 3.6. Assume that A[p ∞ ](F) = 0 or that F itt Λ(Fd )(S(F d )) is principal forany F d . If corank Zp Sel A (F 1 ) Γ(F 1)p ≥ 1 for any Z p -extension F 1 /F contained in F, then˜F itt Λ (S(F)) ⊂ I (where I is the augmentation ideal of Λ).Proof. Recall that I F dis the augmentation ideal of Λ(F d ), that is, the kernel of π F d: Λ(F d ) →Z p . By hypothesis F itt Zp ((Sel A (F 1 ) Γ(F 1)p ) ∨ ) = 0. Thus, since Z p = Λ(F 1 )/I F 1and (Sel A (F 1 ) Γ(F 1)p ) ∨ =S(F 1 )/I F 1S(F 1 ),0 = F itt Zp ((Sel A (F 1 ) Γ(F 1)p ) ∨ ) = π F 1(F itt Λ(F1 )(S(F 1 ))) ,i.e., F itt Λ(F1 )(S(F 1 )) ⊂ Ker π F 1= I F 1.For any Z d p-extension F d take a Z p -extension F 1 contained in F d . Then, by Lemma 3.3,π F dF 1(F itt Λ(Fd )(S(F d ))) ⊆ F itt Λ(F1 )(S(F 1 )) ⊂ I F 1.□
Note that π F d= π F 1◦ π F dF 1. Thereforei.e., F itt Λ(Fd )(S(F d )) ⊂ I F dIWASAWA THEORY OVER FUNCTION FIELDS 11F itt Λ(Fd )(S(F d )) ⊂ I F d⇐⇒ π F dF 1(F itt Λ(Fd )(S(F d ))) ⊂ I F 1,for any Z p -finite extension F d . Finally˜F itt Λ (S(F)) := ⋂ F d(π Fd ) −1 (F itt Λ(Fd )(S(F d ))) ⊂ ⋂ F d(π Fd ) −1 (I F d) ⊂ I .Remark 3.7. From the exact sequenceKer a F1 ↩→ Sel A (F ) pa F1−−→Sel A (F 1 ) Γ(F 1)p ↠ Coker a F1and the fact that Ker a F1 is finite one immediately finds out that the hypothesis on corank Zp Sel A (F 1 ) Γ(F 1)pis satisfied if rank Z A(F ) ≥ 1 or corank Zp Coker a F1 ≥ 1. As already noted, when there isa totally ramified prime of split multiplicative reduction, the second option is very likely tohappen. In the number field case, when F is the cyclotomic Z p -extension and, in some cases,Sel E (F) ∨ p is known to be a torsion module, this is equivalent to saying that T divides a generatorof the characteristic ideal of Sel E (F) ∨ p (i.e., there is an exceptional zero). Note thatall the available constructions of p-adic L-function for our setting require a ramified place ofsplit multiplicative reduction and they are all known to belong to I.4. Modular abelian varieties of GL 2 -typeThe previous sections show how to define the algebraic (p-adic) L-function associated withF/F and an abelian variety A/F under quite general conditions. On the analytic side thereis, of course, the complex Hasse-Weil L-function L(A/F, s), so the problem becomes to relateit to some element in an Iwasawa algebra. In this section we will sketch how this can be doneat least in some cases; in order to keep the paper to a reasonable length, the treatment herewill be very brief.We say that the abelian variety A/F is of GL 2 -type if there is a number field K such that[K : Q] = dim A and K embeds into End F (A) ⊗ Q. In particular, this implies that for anyl ≠ p the Tate module T l A yields a representation of G F in GL 2 (K ⊗ Q l ). The analogousdefinition for A/Q can be found in [40], where it is proved that Serre’s conjecture impliesthat every simple abelian variety of GL 2 -type is isogenous to a simple factor of a modularJacobian. We are going to see that a similar result holds at least partially in our functionfield setting.4.1. Automorphic forms. Let A F denote the ring of adeles of F . By automorphic form forGL 2 we shall mean a function f : GL 2 (A F ) −→ C which factors through GL 2 (F )\GL 2 (A F )/K,where K is some open compact subgroup of GL 2 (A F ); furthermore, f is cuspidal if it satisfiessome additional technical condition (essentially, the annihilation of some Fourier coefficients).A classical procedure associates with such an f a Dirichlet sum L(f, s): see e.g. [49, ChaptersII and III].The C-vector spaces of automorphic and cuspidal forms provide representations of GL 2 (A F ).Besides, they have a natural Q-structure: in particular, the decomposition of the space ofcuspidal forms in irreducible representations of GL 2 (A F ) holds over Q (and hence over anyalgebraically closed field of characteristic zero); see e.g. the discussion in [39, page 218]. Wealso recall that every irreducible automorphic representation π of GL 2 (A F ) is a restrictedtensor product ⊗ ′ vπ v , v varying over the places of F : the π v ’s are representations of GL 2 (F v )and they are called local factors of π.□
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