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.     

Sur les zéros des séries d'Eisenstein dans le demi-plan de Drinfeld

Eisenstein series for GL₂(F_q[T]) of weight q^k − 1 have zeroes in the Drinfeld upper half-plane. Let F be a fundamental domain for the GL₂(A)-action. We determine the number of zeroes in F of these series. Our method is essentially based on an assocíation between Eisenstein series and some functions defined on the edges of the Bruhat-Tits tree.     

The Galois closure of Drinfeld modular towers

In this article we study Drinfeld modular curves X₀(pn) associated to congruence subgroups Γ₀(pn) of GL(2,Fq[T]) where p is a prime of Fq[T]. For n>r>0 we compute the extension degrees and investigate the structure of the Galois closures of the covers X₀(pn)→X₀(pr) and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies.     

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.     

Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.     

On the reduction of a non-torsion point of a Drinfeld module

Let ρ be a Drinfeld Fq[T]-module defined over a global function field K. Let z∈K be a non-torsion point. We prove that for almost all monic elements n∈Fq[T] there exists a place ℘ of K such that n is the “order” of the reduction of z modulo ℘.     

Pesenti-Szpiro inequality for optimal elliptic curves

We study Pesenti-Szpiro inequality in the case of elliptic curves over F_q(t) which occur as subvarieties of Jacobian varieties of Drinfeld modular curves. In general, we obtain an upper-bound on the degrees of minimal discriminants of such elliptic curves in terms of the degrees of their conductors and q. In the special case when the level is prime, we bound the degrees of discriminants only in terms of the degrees of conductors. As a preliminary step in the proof of this latter result we generalize a construction (due to Gekeler and Reversat) of 1-dimensional optimal quotients of Drinfeld Jacobians.     

Sp(2N)-covers for self-contragredient supercuspidal representations of GL(N)

Let F be a non-archimedean local field of odd residual characteristic. Let (J,τ) be a maximal simple type in GLN(F) for the inertial class [GLN(F),π]_GLN(F) of a self-contragredient supercuspidal irreducible representation π of GLN(F). Identify GLN(F) to the standard Siegel Levi subgroup in Sp2N(F). We construct, in Sp2N(F), a type for the inertial class [GLN(F),π]_Sp2N(F), as a Sp2N(F)-cover of (J,τ), strongly related to the GL2N(F)-cover of (J×J,τ⊗τ) in GL2N(F) constructed by Bushnell and Kutzko and which induces to a simple type in GL2N(F). In the process, we show that if τ has positive level, then the maximal simple type (J,τ) may be attached to a simple stratum [A,n,0,β] where the field F[β] is a quadratic extension of F[β²], and to a simple character θ in C(A,0,β) Galois conjugate of its inverse.     

Certain series attached to an even number of elliptic modular forms

For j=1,…,n let f_j(z) and g_j(z) be holomorphic modular forms for such that f_j(z)g_j(z) is a cusp form. We define a series

     

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

     

Minimal models for Drinfeld modules of rank 2 with complex multiplication

We show that some Drinfeld modules of rank 2 on \Bbb F_q[T]{\Bbb F}_q[T] with complex multiplication have global minimal models.     

On modular invariants of a vector and a covector

Let GL ₂(F _q ) be the general linear group over a finite field F _q , V be the 2-dimensional natural representation of GL ₂(F _q ) and V ^* be the dual representation. We denote by \({F_{q}[V\oplus V^{\ast}]^{GL_{2}(F_{q})}}\) the corresponding invariant ring of a vector and a covector for GL ₂(F _q ). In this paper, we prove that \({F_{q}[V\oplus V^{\ast}]^{GL_{2}(F_{q})}}\) is a Gorenstein algebra. This result confirms a special case (n = 2) of the recent conjecture of Bonnafé and Kemper (J Algebra 335:96–112, 2011).     

On the subfields of cyclotomic function fields

Let K = F(T) be the rational function field over a finite field of q elements. For any polynomial f(T) ∈ F [T] with positive degree, denote by Λ _f the torsion points of the Carlitz module for the polynomial ring F[T]. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield M of the cyclotomic function field K(Λ _P ) of degree k over F(T), where P ∈ F[T] is an irreducible polynomial of positive degree and k > 1 is a positive divisor of q ? 1. A formula for the analytic class number for the maximal real subfield M ⁺ of M is also presented. Futhermore, a relative class number formula for ideal class group of M will be given in terms of Artin L-function in this paper.     

A non-standard Fourier expansion for the Drinfeld discriminant function

We find an expansion for the Drinfeld discriminant function indexed by the set of monic polynomials of \mathbb F_q[T]{{\mathbb F}_q[T]} .     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.     

L-functions of twisted Legendre curves

Let K be a global field of char p and let Fq be the algebraic closure of Fp in K. For an elliptic curve E/K with nonconstant j-invariant, the L-function L(T,E/K) is a polynomial in 1+T⋅Z[T]. For any N>1 invertible in K and finite subgroup T⊂E(K) of order N, we compute the mod N reduction of L(T,E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic rank of E/K. We construct infinite families of curves of rank zero when q is an odd prime power such that for some odd prime ?. Our construction depends upon a construction of infinitely many twin-prime pairs (Λ,Λ−1) in Fq[Λ]×Fq[Λ]. We also construct infinitely many quadratic twists with minimal analytic rank, half of which have rank zero and half have (analytic) rank one. In both cases we bound the analytic rank by letting T≅Z/2⊕Z/2 and studying the mod-4 reduction of L(T,E/K).     

The generating function for traces of singular moduli and an application to Borcherds products

Let p be a prime and f(z)=∑ n a(n)q n be a weakly holomorphic modular function for \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}) with a(0)=0. We use Bruinier and Funke’s work to find the generating series of modular traces of f(z) as Jacobi forms. And as an application we construct Borcherds products related to the Hauptmoduln jp2*j_{p^{2}}^{*} for genus zero groups \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}).     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

The sixth moment of the family of Γ<Subscript>1</Subscript>(<Emphasis Type="Italic">q</Emphasis>)-automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper we investigate the sixth moment of the family of L-functions associated to holomorphic modular forms on GL ₂ with respect to a congruence subgroup Γ₁(q). The bound for central values averaged over the family, consistent with the Lindelöf hypothesis, is obtained for prime levels q.