Kloosterman sums and traces of singular moduli

We give a new proof of some identities of Zagier relating traces of singular moduli to the coefficients of certain weakly holomorphic half integral weight modular forms. These identities play a central role in Zagier's work on the infinite product isomorphism introduced by Borcherds. In addition, we derive a simple expression for writing twisted traces of singular moduli as infinite series.     

Exact formulas for traces of singular moduli of higher level modular functions

Zagier proved that the traces of singular values of the classical j-invariant are the Fourier coefficients of a weight 3/2 modular form and Duke provided a new proof of the result by establishing an exact formula for the traces using Niebur's work on a certain class of non-holomorphic modular forms. In this short note, by utilizing Niebur's work again, we generalize Duke's result to exact formulas for traces of singular moduli of higher level modular functions.     

On the congruences of Jacobi forms

In this paper, we study congruence properties of coefficient of Jacobi forms. The result for elliptic modular form case was studied by Sturm (Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York 1987).     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

Symmetric square L-values and dihedral congruences for cusp forms

Let be a prime, and k=(p+1)/2. In this paper we prove that two things happen if and only if the class number . One is the non-integrality at p of a certain trace of normalised critical values of symmetric square L-functions, of cuspidal Hecke eigenforms of level one and weight k. The other is the existence of such a form g whose Hecke eigenvalues satisfy “dihedral” congruences modulo a divisor of p (e.g. p=23, k=12, g=Δ). We use the Bloch-Kato conjecture to link these two phenomena, using the Galois interpretation of the congruences to produce global torsion elements which contribute to the denominator of the conjectural formula for an L-value. When , the trace turns out always to be a p-adic unit.     

Iwasawa invariants for the False-Tate extension and congruences between modular forms

For an ordinary prime p?3, we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q(μp) whose Galois group is G≅Zp?Zp. For Selmer groups defined over the cyclotomic Zp-extension of Q(μp), we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.     

Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

We relate the existence of Frobenius morphisms into the Jacobians of Drinfeld modular curves to the existence of congruences between cusp forms.     

Congruences for modular forms of weights two and four

We prove a conjecture of Calegari and Stein regarding mod p congruences between modular forms of weight four and the derivatives of modular forms of weight two.     

Structure of digraphs associated with quadratic congruences with composite moduli

We assign to each positive integer n a digraph G(n) whose set of vertices is H={0,1,…,n-1} and for which there exists a directed edge from a∈H to b∈H if . Associated with G(n) are two disjoint subdigraphs: G₁(n) and G₂(n) whose union is G(n). The vertices of G₁(n) correspond to those residues which are relatively prime to n. The structure of G₁(n) is well understood. In this paper, we investigate in detail the structure of G₂(n).     

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

Some curious congruences modulo primes

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence:

     

Weight-dependent congruence properties of modular forms

In this paper, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions, in terms of the weights of modular forms, for a modular form to be non-p-ordinary. As applications of our main theorem we derive a linear relation among coefficients of new forms. Furthermore, congruence relations among special values of Dedekind zeta functions of real quadratic fields are derived.     

p-Adic liftings of the supersingular j-invariants and j-zeros of certain Eisenstein series

Let p>3 be a prime. We consider j-zeros of Eisenstein series Ek of weights k=p−1+Mpa(p²−1) with M,a?0 as elements of . If M=0, the j-zeros of Ep−1 belong to Qp(ζp²−1) by Hensel's lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of Ek such that ordp(r−u)>a. Applications of this result are considered. The proof is based on the techniques of formal groups.     

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes études Scientifiques for their hospitality.     

On the Fourier coefficients of Hilbert-Maass wave forms of half integral weight over arbitrary algebraic number fields

The purpose of this paper is to derive a generalization of Shimura's results concerning Fourier coefficients of Hilbert modular forms of half integral weight over total real number fields in the case of Hilbert-Maass wave forms over algebraic number fields by following the Shimura's method. Employing theta functions, we shall construct the Shimura correspondence Ψ_τ from Hilbert-Maass wave forms f of half integral weight over algebraic number fields to Hilbert-Maass wave forms of integral weight over algebraic number fields. We shall determine explicitly the Fourier coefficients of in terms of these f. Moreover, under some assumptions about f concerning the multiplicity one theorem with respect to Hecke operators, we shall establish an explicit connection between the square of Fourier coefficients of f and the central value of quadratic twisted L-series associated with the image of f.     

Note on divisor function for quaternion algebras

Let a be an integral ideal in a quaternion algebra U over rational numbers Q which ramifies precisely at p and ∞, and d(a) be its divisor function. Recently, Kim and Zhang proved a quaternion analogue of the classical formula , and then established an asymptotic formula for the sum . In this note we improved the upper bound for the error term in the asymptotic formula, and then considered the quaternion analogue of another well-known formula .     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.