Abstract
Consider the Drinfeld modular curve \(X_{0}(\mathfrak {p})\) for Open image in new window a prime ideal of \(\mathbb {F}_{q}[T]\). It was previously known that if j is the j-invariant of a Weierstrass point of \(X_{0}(\mathfrak {p})\), then the reduction of j modulo Open image in new window is a supersingular j-invariant. In this paper, we show the converse: Every supersingular j-invariant is the reduction modulo Open image in new window of the j-invariant of a Weierstrass point of \(X_{0}(\mathfrak {p})\).
Highlights
Introduction and statement of resultsGiven a smooth irreducible projective curve of genus g ≥ 2 defined over an algebraically closed field of characteristic 0, we say that a point P on X is a Weierstrass point if there is a nonzero rational function F on X with a pole of order less than or equal to g at P and regular everywhere else
In Section ‘Hyperderivatives and quasimodular forms’, we introduce Drinfeld quasimodular forms and some differentials operators that are needed in the definition of the Drinfeld modular form W (z)
We prove some elementary results concerning the action of these operators on Drinfeld modular forms
Summary
We turn our attention to Drinfeld modular curves and to the family X0(p). The map f → f (z)dz identifies the space of double cusp forms of weight 2 and type 1 for to the space of regular differential forms on X. From this theorem, it follows that the dimension of the space of double cusp forms of weight 2 and type 1 for is g , where g is the genus of the curve X. It follows by a standard argument that all spaces of Drinfeld modular forms of a fixed weight and type for a congruence group are finite dimensional. The space of Drinfeld double cusp forms of weight 2 and type 1 for 0(p) has a basis of forms with integral coefficients.
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