[Paper Review] On the Iwasawa Main conjecture of abelian varieties over function fields

King Fai Lai, Ignazio Longhi+2 more|arXiv (Cornell University)|May 27, 2012

Algebraic Geometry and Number Theory30 references5 citations

TL;DR

This paper proves special cases of the Iwasawa Main Conjecture for abelian varieties over function fields, establishing a p-adic L-function that interpolates twisted Hasse-Weil L-functions and satisfies a characteristic ideal equality linking Selmer groups of an abelian variety and its dual. The key result is a pseudo-isomorphism between the dual Selmer groups of A and its dual A^t, proven via a general algebraic functional equation in Iwasawa theory.

ABSTRACT

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation.

Motivation & Objective

• To establish the Iwasawa Main Conjecture for abelian varieties over function fields in two key cases: constant ordinary abelian varieties over Z_p^d-extensions and semistable abelian varieties over the arithmetic Z_p-extension.
• To prove a general algebraic functional equation for Selmer groups of abelian varieties over global fields, relating the dual Selmer groups of A and its dual A^t.
• To construct a p-adic L-function in the Iwasawa algebra Q(Λ) that interpolates twisted Hasse-Weil L-values at critical points.
• To show that the characteristic ideal of the dual Selmer group of A is equal to that of A^t under potentially ordinary reduction, via a pseudo-isomorphism.

Proposed method

• Uses a pseudo-isomorphism between the dual Selmer groups of an abelian variety A and its dual A^t, derived from a general algebraic functional equation in Iwasawa theory.
• Applies controlled Γ-systems and pairing theory to relate Selmer groups to syntomic cohomology and Galois cohomology.
• Employs Frobenius actions and Iwasawa theory for Z_p^d-extensions to analyze the structure of Selmer groups.
• Utilizes the Iwasawa algebra Λ = Z_p[[Γ]] and its fraction field Q(Λ) to define and study p-adic L-functions.
• Constructs a p-adic L-function L_A/L ∈ Q(Λ) that interpolates twisted L-values L_T(A,ω,1) for all continuous characters ω of Γ.
• Applies the anticyclotomic involution λ ↦ λ^# on Λ to relate the characteristic ideals of Selmer groups of A and A^t.

Experimental results

Research questions

• RQ1 Does the Iwasawa Main Conjecture hold for constant ordinary abelian varieties over Z_p^d-extensions of function fields?
• RQ2 Is there a p-adic L-function in the Iwasawa algebra that interpolates twisted Hasse-Weil L-values for abelian varieties over function fields?
• RQ3 How are the dual Selmer groups of an abelian variety A and its dual A^t related in the Iwasawa module setting?
• RQ4 Can the characteristic ideal of the dual Selmer group of A be related to that of A^t via a functional equation in the Iwasawa algebra?
• RQ5 What is the precise relationship between the p-adic L-function and the characteristic element of the Selmer group in the function field setting?

Key findings

• The paper constructs a p-adic L-function L_A/L ∈ Q(Λ) such that for any continuous character ω: Γ → C^×, the value ω(L_A/L) is proportional to L_T(A,ω,1), with an explicit fudge factor *_{A,T,ω} that is 1 in the semistable case.
• In the case of constant ordinary abelian varieties, the p-adic L-function satisfies L_A/L ≡ ⋆_{A,L} · c_{A/L} mod Λ^× with ⋆_{A,L} = 1, so the p-adic L-function is a generator of the characteristic ideal.
• For semistable abelian varieties over the arithmetic Z_p-extension, the p-adic L-function interpolates the twisted L-values with no fudge factor, and the characteristic element c_{A/L} generates the same ideal as the p-adic L-function.
• The dual Selmer group X_p(A/L) is pseudo-isomorphic to X_p(A^t/L) under potentially ordinary reduction at ramified places, i.e., X_p(A/L) ∼ X_p(A/L)^# ∼ X_p(A^t/L), proving a functional equation for Selmer groups.
• The characteristic ideal of X_p(A/L) satisfies χ(X_p(A/L)) = χ(X_p(A/L))^# = χ(X_p(A^t/L)), which holds even when the Selmer group is not torsion.
• In a non-torsion example, the p-adic L-function vanishes identically, consistent with the expectation that the characteristic ideal is zero.