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.     

A remark on the normalization of the standard zeta values for Siegel modular forms

In (Katsurada in Math. Z. 259:97–111, 2008), we gave a certain type of normalization of the standard zeta values for Siegel modular forms, and considered the relationship between such values and congruence of cuspidal Hecke eigenforms. In this paper we give more reasonable normalization for such values and improve our previous result.     

Rational decomposition of modular forms

We prove explicit formulas decomposing cusp forms of even weight for the modular group, in terms of generators having rational periods, and in terms of generators having rational Fourier coefficients. Using the Shimura correspondence, we also give a decomposition of Hecke cusp forms of half integral weight k+1/2 with k even in terms of forms with rational Fourier coefficients, given by Rankin–Cohen brackets of theta series with Eisenstein series.     

Harmonic Maass forms and periods

According to Waldspurger’s theorem, the coefficients of half-integral weight eigenforms are given by central critical values of twisted Hecke $L$ -functions, and therefore by periods. Here we prove that the coefficients of the holomorphic parts of weight $1/2$ harmonic Maass forms are determined by periods of algebraic differentials of the third kind on modular and elliptic curves.     

On the analogue of Weil’s converse theorem for Jacobi forms and their lift to half-integral weight modular forms

We generalize Weil’s converse theorem to Jacobi cusp forms of weight k, index m and Dirichlet character χ over the group Γ ₀(N)⋉ℤ². Then two applications of this result are given; we generalize a construction of Jacobi forms due to Skogman and present a new proof for several known lifts of such Jacobi forms to half-integral weight modular forms.     

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.

     

Differential eigenforms

The aim of this paper is to show how differential characters of Abelian varieties (in the sense of [A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995) 309-340]) can be used to construct differential modular forms of weight 0 and order 2 (in the sense of [A. Buium, Differential modular forms, Crelle J. 520 (2000) 95-167]) which are eigenvectors of Hecke operators. These differential modular forms will have “essentially the same” eigenvalues as certain classical complex eigenforms of weight 2 (and order 0).     

Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N, with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.     

A generalization of Cohen- Eisenstein series and Shimura liftings and some congruences between Cusp forms and Eisenstein series

In this paper we introduce some modular forms of half-integral weight on congruence group Г_o(4N) withN an odd positive integer which can be viewed as a natural generalization of Cohen-Eisenstein series. Using these series, we can prove that the restriction of Shimura lifting on Eisenstein spaceE _k+1/2 ⁺ (4N,χl) gives an isomorphism fromE _k+1/2 ⁺ (4N,χl) toE _2k(N). We consider some congruence relationships between modular forms in use of Shimura lifting.     

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.     

Some computational results on Hecke eigenvalues of modular forms on a unitary group

In this paper we present some computational results on Hecke eigenforms and eigenvalues for a unitary group in three variables. Our results are based on the work of Shiga [SHig], Holzapfel [Holz1,Holz2] and Feustel ]Feustel] which gives in a special case a generating system for the ring of (holomorphic) modular forms consisting of powers of theta constants. We compute all Hecke eigenforms in this ring for weights up to 12 and for each eigenform the first Hecke eigenvalues. Received: 25 July 1997 / Revised version: 7 January 1998     

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salié sums, and a well known reformulation of Salié sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincaré series in Kohnen’s Γ₀(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.     

Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables

Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq) to the space of classical holomorphic Siegel modular forms of weight (p +q)/2 and genusn (resp. holomorphic hermitian modular forms of weightp +q and genusn). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of η in terms of periods of η over certain totally geodesic cycles—generalizing Shintani’s solution [21] of a conjecture of Shimura. We then choose η to be the Poincaré dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in theholomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher’s Theorem we may now conclude that all such coefficients are zero. Partially supported by NSF Grant # MCS-82-01660. Partially supported by NSF Grant # DMS-85-01742.     

Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.     

On generalized modular forms supported on cuspidal and elliptic points

Suppose N∈{13,17,19,21,26,29,31,34,39,41,49,50}. In this paper, we extend previous results of Kohnen–Mason (On the canonical decomposition of generalized modular functions, 2010) to prove that generalized modular forms for Γ ₀(N) with rational Fourier expansions whose divisors are supported only at the cusps and at the elliptic points are actually classical modular forms. We discuss possible limitations to this extension and pose questions about possible zeroes for modular forms of prime level.     

A certain Dirichlet series of Rankin-Selberg type associated with the Ikeda lifting

We consider a certain Dirichlet series of Rankin-Selberg type associated with two Siegel cusp forms of the same integral weight with respect to Spn(Z). In particular, we give an explicit formula for the Dirichlet series associated with the Ikeda lifting of cuspidal Hecke eigenforms with respect to SL₂(Z). We also comment on a contribution to the Ikeda's conjecture on the period of the lifting.     

L-functions for Jacobi forms and the basis problem

This paper deals with Jacobi forms Φ on ?×ℂ. The Rankin–Selberg doubling method is employed to study properties of the standard L-function of Hecke–Jacobi eigenforms. It is shown that every analytic Klingen–Jacobi Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. Hecke–Jacobi cusp eigenforms of weight k > 4 and k≡ 0 mod 4 can written explicitly as a linear combination of theta series. Finally the basis problem of Jacobi forms of square-free index is solved. Received: 12 March 2000 / Revised version: 17 September 2001     

On geometric density of Hecke eigenvalues for certain cusp forms

In this paper we prove certain density results for Hecke eigenvalues as well as we give estimates on the length of modules for Hecke algebra acting on the cusp forms constructed out of Poincaré series for a semisimple group G over a number field k. The cusp forms discusses here are taken from Muić (Math Ann 343:207–227, 2009) and they generalize usual cuspidal modular forms S _k (Γ) of weight k ≥ 3 for a Fuchsian group Γ (Muić, in On the cuspidal modular forms for the Fuchsian groups of the first kind).     

Eichler–Shimura theory for mock modular forms

We use mock modular forms to compute generating functions for the critical values of modular $L$ -functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler–Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler–Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on $M_{k}^{!}$ in terms of periods.