Integral Weight Modular Forms Research Articles

Modular forms and period polynomials

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for $\Gamma_1(N)$: an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

Logarithmic vector-valued modular forms

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $\rho(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincar\'e series associated to $\rho$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\mathcal{H}(\rho)=\oplus_k \mathcal{H}(k, \rho)$ of all holomorphic logarithmic vector-valued modular forms associated to $\rho$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.

Zagier duality for harmonic weak Maass forms of integral weight

We show the existence of "Zagier duality" between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass forms as developed in recent works by Bruinier, Funke, Ono, and Rhoades. Concerning the isomorphism between the spaces of scalar and vector valued harmonic weak Maass forms of integral weight, "Zagier duality" between scalar valued ones is derived.

Level lowering for half-integral weight modular forms

In this paper we provide a level lowering result for half-integral weight modular forms. The main ingredients are the Shimura map from half-integral weight modular forms to integral weight modular forms along with a level lowering result for integral weight modular forms due to Ribet. It is necessary to keep track of the parity of the weight as well as the character involved so that one can apply the Shintani lift to go back to a half-integral weight modular form and establish the result.

2-Dimensional vector-valued modular forms

We study the graded space ℋ(ρ) of holomorphic vector-valued modular forms of integral weight associated to a 2-dimensional irreducible representation ρ of SL(2,ℤ). When ρ(T) is unitary, a complete description of ℋ(ρ) is given: the Poincaré series is calculated and it is shown that ℋ(ρ) is a free module of rank 2 over the ring of (classical) holomorphic modular forms on SL(2,ℤ).

Differential automorphisms for modular forms on Γ0(4)

If \(k\in\frac{1}{2}\mathbb{N}\) , then let Mk(Γ0(4)) be the usual space of half integral weight modular forms. Ono constructed differential endomorphisms of Mk(Γ0(4)) by using the usual differential operator. Here we construct a similar set of differential endomorphisms using a linear combination of the differential operator and the quasi-modular forms E2, E2|V2, and E2|V4. We compute a full set of eigenforms with eigenvalues, and we prove that these endomorphisms are in fact automorphisms.

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.

Existence of certain fundamental discriminants and class numbers of real quadratic fields

Let D be the fundamental discriminant of the quadratic field Q( √ D), h(D) its class number, and χD := ( D · ) the usual Kronecker character. Let p be prime, Zp the ring of p-adic integers, and λp(Q( √ D)) the Iwasawa λ-invariant of the cyclotomic Zp-extension of Q( √ D). Let Rp(D) denote the p-adic regulator of Q( √ D), and | · |p denote the usual multiplicative p-adic valuation normalized so that |p|p = 1 p . In [9], by applying Sturm’s theorem on the congruence of modular forms to Cohen’s half integral weight modular forms, Ono proved the following theorem.

A certain relation between Jacobi forms of half integral weight and siegel modular forms of integral weight

A certain relation between Jacobi forms of half integral weight and siegel modular forms of integral weight

Toric modular forms and nonvanishing of L-functions

In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) 6= 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

On Shimura, Shintani and Eichler-Zagier correspondences

In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.

On Shimura lifting of modular forms

0. Let k,NeN, 4\N. Let M/c+\/2{N,Xo) denote the space of modular forms for Tq(N) of weight k+l/2 with a character xo (modJV). Lifting maps of cusp forms in M^+i^iN^Xo) to modular forms of integral weight was firststudied by Shimura [7] and later by Niwa [4]. The domain of the map is extended to Mk+\ii(N,Xo) by van Asch [1]in case that Xo is real and N = 4p for p prime, and by Pei [5] in case that Xo is real and N/4 is square-free.In the present paper we consider the liftingmap without any condition on N and /0, and extend the domain of the map to Mk+\/2(N,Xo) f°rk >2. To show the assertion, we take some specificmodular forms in Mk+i/2{N,Xo) which together with cusp forms, span M^+1/2(Ar,/0). Further we construct their liftingsexplicitly.It proves our main result.It may be expected to have further application to study of special values of L-series of Hecke eigen cusp forms, as in Zagier [9],Kohnen-Zagier [3] where the liftingof some particular modular forms plavs an important role.

On the Ramanujan-Petersson conjecture for modular forms of half-integral weight

It is shown that a “character twist” of a growth estimate for certain weighted infinite sums of Kloosterman sums which is equivalent to the Ramanujan-Petersson conjecture for modular forms of half-integral weight, can easily be proved using Deligne’s theorem (previously the Ramanujan-Petersson conjecture for modular forms of integral weight).

On twisting operators and newforms of half-integral weight

The theory of newforms is very important and useful for arithmetical study of modular forms of integral weight. It is natural to try to extend this theory into the case of modular forms of half-integral weight Until now, several authors have attempted to find a theory of newforms of half-integral weight (cf. [She], [N], [K], [M-R-V], [She-W]). But complete results have not been obtained yet.

Supercuspidal Duality for the Two-Fold Cover of SL 2 and the Split O 3

Introduction. As first introduced the Shimura-Shintani-Niwa correspondence was a correspondence between homomorphic cusp forms of half-integral weight and certain modular forms of integral weight. Various authors have studied this correspondence in a representation-theoretic framework. Gelbart's seminal work [G1] provides some tools for translation and miscellaneous results. Using Whittaker models and L-functions, Gelbart and Piatetski-Shapiro [GPS1], [GPS2], obtained a generalized correspondence between automorphic cuspidal representations of GL2, the metaplectic group, and automorphic representations of GL2. Flicker [F] used a variation on the Selberg trace formula to extend the work of Gelbart and Piatetski-Shapiro. Finally, Waldspurger [W] returned to Whittaker models andL-functions to obtain a correspondence between automorphic cuspidal representations of SL2, the nontrivial two-fold cover of SL2, and automorphic representations of PGL2. Our more modest results complement the above and are obtained by purely algebraic means. We now attempt a brief summary.

Ternary quadratic forms and Shimura’s correspondence

In his paper [11] Shimura defined a correspondence between modular forms of half integral weight and modular forms of integral weight. To each pair (t, f(z)), consisting of a square-free integert≥ 1 and a cusp formof weightk/2 (kodd, ≥ 3), levelN(divisible by 4) and characterϰ, he associated a certain functionf(t)(z) (Ft(z) in Shimura’s notation).

Modular forms of half integral weight and the integral of certain theta-functions

Recently G. Shimura [1] constructed modular forms of integral weight from the forms of half integral weight. His construction is rather indirect. Indeed, he proved that the Dirichlet series, obtained from a form of half integral weight, multiplied by a certain L-function, corresponds to a modular form of an integral weight by means of the characterization of modular forms due to Weil.