Drinfeld modular polynomials in higher rank

We study modular polynomials classifying cyclic isogenies between Drinfeld modules of arbitrary rank over the ring Fq[T]. We derive bounds for the coefficients of these polynomials, and compute some explicit examples in the case where q=2, the rank is 3 and the isogenies have degree T.     

Coefficients of Drinfeld modular forms and Hecke operators

Consider the space of Drinfeld modular forms of fixed weight and type for Γ₀(n)⊂GL₂(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b₁. We give elements in the Hecke annihilator of b₁. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.     

Special values of Drinfeld modular forms and algebraic independence

Let A be a polynomial ring in one variable over a finite field and k be its fraction field. Let f be a Drinfeld modular form of nonzero weight for a congruence subgroup of GL₂(A) so that the coefficients of the q _∞-expansion of f are algebraic over k. We consider n CM points α ₁, . . . , α _n on the Drinfeld upper half plane for which the quadratic fields k(α ₁), . . . , k(α _n ) are pairwise distinct. Suppose that f is non-vanishing at these n points. Then we prove that f(α ₁), . . . , f(α _n ) are algebraically independent over k.     

Zeta elements in the K-theory of Drinfeld modular varieties

Beilinson (Contemp Math 55:1?C34, 1986) constructs special elements in the second K-group of an elliptic modular curve, and shows that the image under the regulator map is related to the special values of the L-functions of elliptic modular forms. In this paper, we give an analogue of this result in the context of Drinfeld modular varieties.     

The André–Oort conjecture for Drinfeld modular varieties

We state an analogue of the André–Oort conjecture for subvarieties of Drinfeld modular varieties, and prove it in two special cases. To cite this article: F. Breuer, C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

Toric varieties and modular forms

Let N⊂ℝ^r be a lattice, and let deg:N→ℂ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N,deg) determines a function f:ℌ→ℂ that is a holomorphic modular form of weight r for the congruence subgroup Γ₁(l). Moreover, by considering all possible pairs (N ,deg), we obtain a natural subring ? (l) of modular forms with respect to Γ₁ (l). We construct an explicit set of generators for ? (l), and show that ? (l) is stable under the action of the Hecke operators. Finally, we relate ? (l) to the Hirzebruch elliptic genera that are modular with respect to Γ₁ (l). Oblatum 22-IX-1999 & 18-X-2000?Published online: 5 March 2001     

Some formulas for the coefficients of Drinfeld modular forms

We obtain some formulas for t-expansion coefficients of meromorphic Drinfeld modular forms for GL₂(F_q[T]). Let j(z) be the Drinfeld modular invariant. As an application we show that the values of j(z) at points in the divisor of Drinfeld modular forms for GL₂(F_q[T]) are algebraic over F_q(T).     

The zeros of certain weakly holomorphic Drinfeld modular forms

Duke and Jenkins (Pure Appl Math Q 4(4):1327–1340, 2008) constructed a canonical basis for the space of weakly holomorphic modular forms for \({{\rm SL}_2(\mathbb{Z})}\) and investigated the zeros of the basis elements. In this paper we give an analogy in the Drinfeld setting of the result given by Duke and Jenkins (Pure Appl Math Q 4(4):1327–1340, 2008).     

Linear relations and congruences for the coefficients of Drinfeld modular forms

We find congruences for the t-expansion coefficients of Drinfeld modular forms for . We give generalized analogies of Siegel’s classical observation on SL ₂(ℤ) by determining all the linear relations among the initial t-expansion coefficients of Drinfeld modular forms for . As a consequence spaces M _k ⁰ are identified, in which there are congruences for the s-expansion coefficients. This work was supported by KOSEF R01-2006-000-10320-0 and by the Korea Research Foundation Grant (KRF-2005-214-M01-2005-000-10100-0)     

Action of Hecke operators on two distinguished Drinfeld modular forms

We study the action of Hecke operators on certain non-standard Fourier expansions for the Drinfeld-Eisenstein series E _q-1 and the Drinfeld discriminant function Δ, and we find an equation which “explains” an old result of D. Goss: these two distinct modular forms have the same eigenvalues.     

On hypercomplexifying real forms of arbitrary rank

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary Clifford algebras, followed by quaternionic vectors as a special case. All results are shown to reduce to the established method of complexifying vector fields. For simplicity, differential forms are used rather than vector notation.     

Compactification des champs de chtoucas de Drinfeld

Another way to construct the compactifications of the stacks of Drinfeld's shtukas introduced by Lafforgue is presented. The method is based on the variation of GIT quotients studied by Thaddeus and Dolgachev–Hu. To cite this article: Tuan Ngo Dac, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Compactification of symmetric varieties

The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a wonderful compactification ofG/H. We prove the existence of such a compactification for arbitraryk. We also prove cohomology vanishing results for line bundles on the compactification. Dedicated to the memory of C. Chevalley     

Zeros of certain Drinfeld modular functions

For every positive integer m, there is a unique Drinfeld modular function, holomorphic on the Drinfeld upper-half plane, jm(z) with the following t-expansion