[Paper Review] $D$-Elliptic Sheaves and Uniformisation

Lenny Taelman|ArXiv.org|Dec 8, 2005

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper establishes an analytic uniformization of moduli spaces of $\mathcal{D}$-elliptic sheaves over function fields by realizing them as quotients of a symmetric space $\Omega^d(\mathbf{C})$ by arithmetic groups, extending Drinfel'd's uniformization of modular curves to a broader class of Shimura-type varieties in positive characteristic.

ABSTRACT

An analytic uniformisation of the varieties of Laumon, Rapoport and Stuhler at the infinite place is presented. This can be seen as the function field analog to the uniformisation of Shimura varieties classifying Abelian varieties with large endomorphism rings.

Motivation & Objective

• To provide an analytic, uniformization-based description of moduli spaces of $\mathcal{D}$-elliptic sheaves, complementing their known algebraic moduli interpretation.
• To extend the classical uniformization of modular curves (via the upper half-plane) to higher-dimensional moduli spaces arising from central simple algebras over function fields.
• To bridge the gap between algebraic moduli spaces (Laumon–Rapoport–Stuhler) and analytic structures by constructing a uniformization at the infinite place.
• To generalize Drinfel'd's uniformization of modular varieties to the case of division algebras and arbitrary level structures.
• To establish a rigid analytic isomorphism between the analytification of the moduli space and a quotient of a symmetric space by an arithmetic group.

Proposed method

• Show that $\mathcal{D}$-elliptic sheaves correspond to a special class of Anderson $A$-motives via a categorical equivalence.
• Prove that these Anderson $A$-motives are uniformizable, i.e., isomorphic to quotients of a vector space by a lattice in $\mathbf{C}^d$.
• Identify the class of lattices arising from uniformization as those corresponding to $\Omega^d(\mathbf{C})$, a symmetric space for $\mathrm{GL}_d(F_\infty)$.
• Construct a map from the moduli space of $\mathcal{D}$-elliptic sheaves to the double coset space $D^* \backslash [\Omega^d(\mathbf{C}) \times D^*(\mathbb{A}_A)/U]$, using the uniformization of $A$-motives.
• Establish a natural isomorphism of rigid analytic spaces between the analytification of the moduli space and the quotient space via a bijection on $\mathbf{C}$-points and reducedness arguments.
• Generalize the result to open and closed subspaces corresponding to free $\mathcal{D}$-elliptic sheaves, using level structures defined by ideals $I$.

Experimental results

Research questions

• RQ1 Can the moduli spaces of $\mathcal{D}$-elliptic sheaves be given an analytic uniformization analogous to the classical uniformization of modular curves?
• RQ2 How does the uniformization of Anderson $A$-motives arising from $\mathcal{D}$-elliptic sheaves relate to the geometry of symmetric spaces over $\mathbf{C}$?
• RQ3 What is the precise analytic structure of the moduli space of $\mathcal{D}$-elliptic sheaves at the infinite place, and how does it compare to the algebraic moduli space?
• RQ4 How does the uniformization generalize to cases with level structures, and what role do arithmetic subgroups play?
• RQ5 In the case of quaternion algebras, how does the uniformization recover known results on Mumford curves and genus computations?

Key findings

• The analytification of the moduli space $\mathcal{E}\ell\ell_{X,\mathcal{D},A}(\mathbf{C})$ is isomorphic to the rigid analytic space $D^* \backslash [\Omega^d(\mathbf{C}) \times D^*(\mathbb{A}_A)/U]$.
• For free $\mathcal{D}$-elliptic sheaves, the analytification of $\mathcal{E}\ell\ell^0_{X,\mathcal{D},I}(\mathbf{C})$ is isomorphic to $G(I)\backslash\Omega^d(\mathbf{C})$, where $G(I)$ is the subgroup of $O_D^*$ reducing to 1 modulo $I$.
• When $D = \mathrm{M}(d,F)$, the result recovers the uniformization of Drinfel'd modular varieties established in [3], confirming consistency with known cases.
• In the case of a quaternion algebra unramified at $\infty$, the moduli space is a smooth, complete curve over $\mathbf{C}$, uniformized as a quotient of $\Omega^2 \subset \mathbb{P}^1(\mathbf{C})$ by a discrete subgroup of $\mathrm{GL}_2(\mathbf{C})$.
• The genus of such a curve is equal to the rank of the abelianization of the uniformizing group, and can be computed via ramification data when a larger group with genus zero quotient is known.
• The uniformization is compatible with the algebraic moduli structure, as the induced map on $\mathbf{C}$-points is a bijection and both spaces are reduced, hence an isomorphism of rigid analytic spaces.