From the Carlitz Exponential to Drinfeld Modular Forms

Abstract

This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest 'Drinfeld modular group' GL 2 (Fq[θ]) also providing some perspectives of development, notably in the direction of the theory of vector modular forms with values in certain ultrametric Banach algebras. Contents 1. Introduction 1 2. Rings and fields 3 3. Drinfeld modules and uniformisation 11 4. The Carlitz module and its exponential 16 5. Topology of the Drinfeld upper-half plane 26 6. Some quotient spaces 37 7. Drinfeld modular forms 43 8. Eisenstein series with values in Banach algebras 50 9. Modular forms with values in Banach algebras 54 References 61 Definition 2.3. A valued field which is locally compact is called a local field. Note that R and C, with their euclidean topology, are locally compact, but not valued. Some authors define local fields as locally compact topological field for a non-discrete topology. Then, they distinguish between the non-Archimedean (or ultrametric) local fields, which are the valued ones, and the Archimedean local fields: R and C. An important property is the following. Any valued local field L of characteristic 0 is isomorphic to a finite extension of the field of p-adic numbers Q p for some p, while any local field L of characteristic p > 0 is isomorphic to a local field F q ((π)), and with q = p e for some integer e > 0. We say that π is an uniformiser. Note that |L × | = |π| Z and |π| < 1. The proof of this result is a not too difficult deduction from the following well known fact: a locally compact topological vector space over a non-trivial locally compact field has finite dimension. 2.2. Valued rings and fields for modular forms. Let C be a smooth, projective, geometrically irreducible curve over F q , together with a closed point ∞ ∈ C. We set R = A := H 0 (C \ {∞}, O C). This is the F q-algebra of the rational functions over C which are regular everywhere except, perhaps, at ∞. The choice of ∞ determines an equivalence class of valuations | • | ∞ on A in the following way. Let d ∞ be the degree of ∞, that is, the degree of the extension F of F q generated by ∞ (which is also equal to the least integer d > 0 such that τ d (∞) = ∞, where τ is the geometric Frobenius endomorphism). Then, for any a ∈ A, the degree deg(a) := dim Fq (A/aA) is a multiple −v ∞ (a)d ∞ of d ∞ and we set |a| ∞ = c −v∞(a) for c > 1, which is easily seen to be a valuation. It is well known that A is an arithmetic Dedekind domain with A × = F ×. In addition

The Unbounded Denominators Conjecture

“…Firstly, the group SL 2 (F q [t]) in function field arithmetic and its attendant theory of Drinfeld-Goss modular forms. See Pellarin [Pel21] for a recent survey of this area. Here, in the analogy with SL 2 (Z) where the congruence kernels of these two arithmetic groups are similarly large, it would be interesting to decide whether the modular forms on a finite index subgroup of SL 2 (F q [t]) that have (up to a F q (t) × scalar multiple) a u-expansion [Pel21, § 4.7.1] with coefficients in A = F q [t] are likewise the congruence modular forms.…”