Duality of Drinfeld modules and \wp-adic properties of Drinfeld modular forms

2017/06/23 by Shin Hattori, Hattori, Shin · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.1706.07645

Abstract

Let p be a rational prime and q a power of p. Let \wp be a monic irreducible polynomial of degree d in F_q[t]. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld modules over F_q[t] and, using it, develop a geometric theory of \wp-adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show that for Drinfeld modular forms with congruent Fourier coefficients at ∞ modulo \wpⁿ, their weights are also congruent modulo (q^d-1)p^{\lceil log_p(n)\rceil}, and that Drinfeld modular forms of level Γ₁(\mathfrakn)∩ Γ₀(\wp), weight k and type m are \wp-adic Drinfeld modular forms for any tame level \mathfrakn with a prime factor of degree prime to q-1.

Cited by

• Higher Hida theory for Drinfeld modular curves
• Arithmetic Satake compactifications and algebraic Drinfeld modular forms
• An explicit form of the canonical submodule of a Drinfeld module

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