[Paper Review] Anderson T-motives and abelian varieties with MIQF: results coming from an analogy

A. Grishkov, Dmitry Logachev|arXiv (Cornell University)|Jul 27, 2009

Algebraic Geometry and Number Theory3 references4 citations

TL;DR

This paper establishes an analogy between Anderson T-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF), leading to two key results: (1) a characterization of MIQF abelian varieties of dimension $ r $ and signature $ (n, r-n) $ via $ r $-dimensional $ O_K $-submodules of $ \mathbb{C}^n $, and (2) a construction of the $ k $-th exterior power of MIQF abelian varieties when $ n=1 $, defined analogously to exterior powers of T-motives. The construction is shown to yield new MIQF abelian varieties with specified signature.

ABSTRACT

Analogy between Anderson T-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: 1. A description of abelian varieties with MIQF of dimension $r$ and signature $(n, r-n)$ in terms of "lattices" of dimension $r$ in $\Bbb C^n$; 2. A construction of exterior powers of abelian varieties with MIQF having $n=1$.

Motivation & Objective

• To explore and formalize an analogy between Anderson T-motives and abelian varieties with MIQF, particularly in the context of uniformizable, pure T-motives with $ N=0 $.
• To provide a lattice-theoretic description of abelian varieties with MIQF of dimension $ r $ and signature $ (n, r-n) $, analogous to the lattice construction for T-motives.
• To define and construct the $ k $-th exterior power of an abelian variety with MIQF when $ n=1 $, mirroring the known construction for T-motives.
• To show that this construction yields a new abelian variety with MIQF, and to relate it to Shimura varieties via the map $ GU(1,r-1) \to GU\left(\binom{r-1}{k-1}, \binom{r-1}{k}\right) $.

Proposed method

• Use of the analogy between T-motives and MIQF abelian varieties to transfer constructions from the function field setting to the number field setting.
• Construction of a lattice $ L(A) $ as an $ r $-dimensional $ O_K $-submodule of $ \mathbb{C}^n $, not $ \mathbb{C}^r $, via the map $ \alpha $, generalizing the T-motive lattice $ L(M) \subset \mathbb{C}_\infty^n $.
• Definition of the exterior power $ \lambda^k(A) $ via the $ k $-th exterior power of the exact sequence $ 0 \to \ker \alpha \to L \otimes_K \mathbb{C} \to \mathbb{C}^n \to 0 $, with induced Hermitian form $ \lambda^k(H_L) $.
• Verification that $ (-1)^k \lambda^k(H_L) $ is positive definite on $ \lambda^k(\ker \alpha) $ when $ n=1 $, ensuring the existence of a well-defined abelian variety.
• Use of the map $ \gamma \mapsto \lambda^k(\gamma) $ to define a morphism $ GU(1,r-1) \to GU\left(\binom{r-1}{k-1}, \binom{r-1}{k}\right) $, linking the construction to Shimura varieties.
• Proof that the two constructions (lattice description and exterior power) are inverse to each other, establishing a bijection up to isogeny.

Experimental results

Research questions

• RQ1 Can abelian varieties with MIQF of signature $ (n,r-n) $ be described analogously to T-motives via $ r $-dimensional $ O_K $-submodules of $ \mathbb{C}^n $, rather than $ \mathbb{C}^r $?
• RQ2 Is there a well-defined construction of the $ k $-th exterior power of an abelian variety with MIQF when $ n=1 $, analogous to that for T-motives?
• RQ3 What is the signature of the $ k $-th exterior power of an MIQF abelian variety of dimension $ r $ and signature $ (1,r-1) $?
• RQ4 Can this construction be extended to a morphism between Shimura varieties $ GU(1,r-1) \to GU\left(\binom{r-1}{k-1}, \binom{r-1}{k}\right) $?
• RQ5 Why does the construction fail for $ n \neq 1, r-1 $, and what prevents the definition of symmetric or tensor powers in this context?

Key findings

• The paper constructs an $ r $-dimensional $ O_K $-submodule of $ \mathbb{C}^n $, denoted $ \gamma(L) $, which serves as a lattice for an abelian variety with MIQF of signature $ (n,r-n) $, generalizing the T-motive lattice $ L(M) \subset \mathbb{C}_\infty^n $.
• For $ n=1 $, the $ k $-th exterior power $ \lambda^k(A) $ of an MIQF abelian variety $ A $ of dimension $ r $ is shown to exist and defines a new abelian variety with MIQF of signature $ \binom{r-1}{k-1}, \binom{r-1}{k} $.
• The Hermitian form $ \lambda^k(H_L) $ restricted to $ \lambda^k(\ker \alpha) $ is positive definite when $ n=1 $, ensuring the existence of a well-defined abelian variety structure.
• The construction of $ \lambda^k(A) $ is shown to be compatible with the theory of Shimura varieties, with the associated map $ GU(1,r-1) \to GU\left(\binom{r-1}{k-1}, \binom{r-1}{k}\right) $ arising naturally from the exterior power of the Galois representation.
• The construction is inverse to the lattice description, establishing a one-to-one correspondence between such MIQF abelian varieties and the corresponding $ O_K $-submodules of $ \mathbb{C}^n $, up to isogeny.
• The method fails for $ n \neq 1, r-1 $, and symmetric or tensor powers cannot be defined in the same way, highlighting the special role of $ n=1 $ in this analogy.