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).     

Restricting Hecke-Siegel operators to Jacobi modular forms

We compute the action of Hecke operators on Jacobi forms of “Siegel degree” n and m×m index M, provided 1?j?n−m. We find they are restrictions of Hecke operators on Siegel modular forms, and we compute their action on Fourier coefficients. Then we restrict the Hecke-Siegel operators T(p), Tj(p²) (n−m<j?n) to Jacobi forms of Siegel degree n, compute their action on Fourier coefficients and on indices, and produce lifts from Jacobi forms of index M to Jacobi forms of index M^′ where detM|detM^′. Finally, we present an explicit choice of matrices for the action of the Hecke operators on Siegel modular forms, and for their restrictions to Jacobi modular forms.     

Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its k-th power form the space of (algebraic) Drinfeld modular forms of weight k. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.     

Compactified Jacobians and q,t-Catalan numbers, II

We continue the study of the rational-slope generalized q,t-Catalan numbers c _m,n (q,t). We describe generalizations of the bijective constructions of J. Haglund and N. Loehr and use them to prove a weak symmetry property c _m,n (q,1)=c _m,n (1,q) for m=kn±1. We give a bijective proof of the full symmetry c _m,n (q,t)=c _m,n (t,q) for min(m,n)≤3. As a corollary of these combinatorial constructions, we give a simple formula for the Poincaré polynomials of compactified Jacobians of plane curve singularities x ^kn±1=y ⁿ . We also give a geometric interpretation of a relation between rational-slope Catalan numbers and the theory of (m,n)-cores discovered by J. Anderson.     

On a sum involving Fourier coefficients of cusp forms

We improve the existing upper bound for the quantity |∑ _n⩽x a(n ²)|, where a(n ²) is the n ²th Hecke eigenvalue of a normalized holomorphic cusp form (Hecke eigenform) of the full modular group SL(2, ℤ), whenever the weight of the original holomorphic cusp form (Hecke eigenform) lies in a certain range. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 565–583, October–December, 2006.     

Commutator subgroups of the power subgroups of?some Hecke groups

Let q??3 be a prime and let H(?? _q ) be the Hecke group associated to q. Let m be a positive integer and H ^m (?? _q ) be the mth power subgroup of H(?? _q ). In this work, we study the commutator subgroups of the power subgroups H ^m (?? _q ) of H(?? _q ). Then, we give the derived series for all triangle groups of the form (0;2,q,n) for n a positive integer, since there is a nice connection between the signatures of the subgroups we studied and the signatures of these derived series.     

Products of Hecke eigenforms

Let f(z) and g(z) be Hecke eigenforms for Γ₀(p), where p is a prime. If both f(z) and g(z) are non-cuspidal forms and p?7, then the product is a Hecke eigenform only if it comes trivially from a level 1 solution. If g(z) is a cuspform and p?5, then in addition to the level 1 solutions, there are 8 new cases where the product of Hecke eigenforms is a Hecke eigenform.     

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.     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.     

A proof of the q,t-Catalan positivity conjecture

We present here a proof that a certain rational function C_n(q,t) which has come to be known as the “q,t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that C_n(q,t) is the Hilbert series of the diagonal harmonic alternants in the variables (x₁,x₂,…,x_n;y₁,y₂,…,y_n). Since C_n(q,t) evaluates to the Catalan number at t=q=1, it has also been an open problem to find a pair of statistics a(π),b(π) on Dyck paths π in the n×n square yielding C_n(q,t)=∑_πt^a(π)q^b(π). Our proof is based on a recursion for C_n(q,t) suggested by a pair of statistics a(π),b(π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363).     

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.     

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)     

Hecke operators on Drinfeld cusp forms

In this paper, we study the Drinfeld cusp forms for Γ₁(T) and Γ(T) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Γ₁(T) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for Γ₁(T) of large weights, and not for Γ(T) even of small weights. The Hecke eigenvalues on cusp forms for Γ(T) with small weights are determined and the eigenspaces characterized.     

Automorphic forms for elliptic function fields

Let F be the function field of an elliptic curve X over ${\mathbb{F}_q}$ . In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over F. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of ${\mathbb{P}^1}$ -bundles on X. This allows a purely geometric approach, which involves, amongst others, a classification of the ${\mathbb{P}^1}$ -bundles on X. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial ${\mathbb{P}^1}$ -bundle. Further, we determine the space of unramified F′-toroidal automorphic forms where F′ is the quadratic constant field extension of F. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke L-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series E( · , s) where s?+?1/2 is a zero of the zeta function of X—with one possible exception in the case that q is even and the class number h equals q?+?1.     

Ramanujan-type congruences for certain generating functions

For nonnegative integers a, b, the function d _a,b (n) is defined in terms of the q-series $\sum_{n=0}^\infty d_{a,b}(n)q^n=\prod_{n=1}^\infty{(1-q^{ an})^b}/{(1-q^n)}$ . We establish some Ramanujan-type congruences for d _a,b (n) by the theory of modular forms with complex multiplication. As consequences, we generalize the famous Ramanujan congruences for the partition function p(n) modulo 5, 7, and 11.     

Uniform estimates for sums of Fourier coefficients of cusp forms

Let k be an even positive integer and f a holomorphic Hecke eigenform of weight k with respect to the full modular group SL(2, ?). Let c _n be the nth coefficient of the symmetric square L-function associated to f. We study the uniform bound for the sum C(x) = Σ _n≦x c _n with respect to the weight k and establish that $$ C(x) = \sum\limits_{n \leqq x} {c_n } \ll x^{\tfrac{3} {5}} (\log x)^{\tfrac{{22}} {5}} + k^{\tfrac{3} {2}} (\log x)^5 $$ . Other similar results are also established.     

On plane models for Drinfeld modular curves

In this work, we find plane models for certain Drinfeld modular curves X₀(n) which have better properties than the plane models derived from the usual Drinfeld modular equations. As an application, we construct ring class fields over imaginary quadratic fields by using singular values of generators of the function field of X₀(n).     

Rank varieties for Hopf algebras

We construct rank varieties for the Drinfeld double of the Taft algebra Λ_n and for u_q(sl₂). For the Drinfeld double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of Λ-modules whenever Λ is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext^∗(M,M) is finitely generated over the cohomology ring of the Drinfeld double for any finitely generated module M.     

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.     

Extremal sizes of subspace partitions

A subspace partition Π of V?= V(n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σ _q (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ _q (n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σ _q (n, t) and ρ _q (n, t) for all positive integers n and t. Furthermore, we prove that if n ≥?2t, then the minimum size of a maximal partial t-spread in V(n +?t ?1, q) is σ _q (n, t).