Drinfeld modular curve and Weil pairing

Abstract

In this paper we describe the compactification of the Drinfeld modular curve. This compactification is analogous to the compactification of the classical modular curve given by Katz and Mazur. We show how the Weil pairing on Drinfeld modules that we defined in earlier work gives rise to a map on the Drinfeld modular curve. We introduce the Tate-Drinfeld module and show how this describes the formal neighbourhood of the scheme of cusps of the Drinfeld modular curve.

Explicit Drinfeld moduli schemes and Abhyankar's Generalized Iteration Conjecture

“…The first three statements are essentially due to Drinfel'd [5], who proved this more generally over Spec A but for level structures divisible by two distinct primes. In our situation, the level tn [20], as well as [7] for a very clear exposition of the situation over the quotient field of A. Alternatively, the interested reader is challenged to deduce (i)-(iii) directly from Theorem 2, for example the fact that Spec(RS W,0 ) → Spec(RS V,0 ) isétale follows exactly as in Proposition 1.…”

“…Statement (iv) is essentially due to Anderson [3], see also [20] for details. To prove (v), note that RS W,0 is flat over B, by (i), so RS W,0 injects into RS W,0 ⊗ B F , which is integral by [7,Cor.…”

“…Thus in our case, Drinfeld's proofs give (i), (ii) and (iii) above. See also [20], as well as [7] for a very clear exposition of the situation over the quotient field of A.…”

Perfectoid Drinfeld Modular Forms

“…is a direct calculation on the Tate-Drinfeld module of rank 2[22, Lemma 6.5], that we denote by TD. This is a rank two Drinfeld module over A p[[x]] which reduces modulo x to the Carlitz-Hayes Drinfeld module.…”

“…Anyway, we have that TD[π]/CH[π] is generated by an element of positive x-adic valuation and it is étale as π-divisible module (cf. the explicit calculation of[22, Lemma 6.5] or[18, Lemma 4.4]). Hence the passage from Γ 0 (π…”