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.     

Identities and congruences for Ramanujan’s ω(q)

Recently, the authors constructed generalized Borcherds products where modular forms are given as infinite products arising from weight 1/2 harmonic Maass forms. Here we illustrate the utility of these results in the special case of Ramanujan’s mock theta function ω(q). We obtain identities and congruences modulo 512 involving the coefficients of ω(q).     

Eichler integrals and harmonic weak Maass forms

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, H-groups by employing the theory of supplementary functions introduced and developed by M.I. Knopp and S.Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms.     

APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

Congruences for Ramanujan’s f and ω functions via generalized Borcherds products

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan’s mock theta functions f and ω in detail.     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

ABSTRACT

In this contribution, we establish a calculus of pseudodifferential boundary value problems with Hölder continuous coefficients. It is a generalization of the calculus of pseudodifferential boundary value problems introduced by Boutet de Monvel. We discuss their mapping properties in Bessel potential and certain Besov spaces. Although having non-smooth coefficients and the operator classes being not closed under composition, we will prove that the composition of Green operators a ₁(x, D _x )a ₂(x, D _x ) coincides with a Green operator a(x, D _x ) up to order m ₁ + m ₂ ? Θ, where Θ ∈ (0, τ₂) is arbitrary, a _j (x, ξ) is in C ^{τ _j} (? ⁿ ) w.r.t. x, and m _j is the order of a _j (x, D _x ), j = 1, 2. Moreover, a(x, D _x ) is obtained by the asymptotic expansion formula of the smooth coefficient case leaving out all terms of order less than m ₁ + m ₂ ? Θ. This result is used to construct a parametrix of a uniformly elliptic Green operator a(x, D _x ).     

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H=AS⁺+A^-S^-+B (with S^± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E₊], E_-<E₊, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by , 0?E_-<E₊.Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.     

Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold

We present simple trace formulas for Hecke operators Tk(p) for all p>3 on Sk(Γ₀(3)) and Sk(Γ₀(9)), the spaces of cusp forms of weight k and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves equipped with a 3-torsion point as special values of a Gaussian hypergeometric series over Fq, when . As an application, we use these formulas to provide a simple expression for the Fourier coefficients of η⁸(3z), the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.     

On the existence of search designs with continuous factors

Consider the search linear model defined as follows. Lety(N×1) be a vector ofN observations such that (1) $$E(y) = A_1 \xi _1 + A_2 \xi _2 ,V(y) = \sigma ^2 I_N$$ whereσ ² may or may not be known,A ₁(N × υ ₁) andA ₂(N ×υ ₂) are known matrices, ξ₁(υ ₁ × 1) is unknown and ξ₂(υ ₂ × 1) is partly known in the following sense. We known that at mostk elements of ξ₂ are non zero but we do not know particularly which these nonzero elements are. The problem is to make inferences about the elements of ξ₁ and, furthermore, to search the nonzero elements of ξ₁ and make inferences about them. We wanty to be such that the above problem can be resolved with certainty whenσ ²=0; the underlying design corresponding toy is then called a search design. It has been shown in earlier work that for a search design, we must haveN≧υ ₁+2k. In this paper, we consider the special case of search linear models, when the object of the experiment is to fit an appropriate response surface. We establish a basic result, namely, that when the true response surface is representable by a polynomial, then search designs exist for whichN=υ ₁+2k, irrespective of the value ofυ ₂.     

On multiple sum and product sets of finite sets of integers

Let $\text{A?}\text{Z}$ be a finite set of integers of cardinality |A|=N?2. Given a positive integer k, denote kA (resp. A^(k)) the set of all sums (resp. products) of k elements of A. We prove that for all b>1, there exists k=k(b) such that max(|kA|,|A^(k)|)>N^b. This answers affirmably questions raised in Erd?s and Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) and recently, by S. Konjagin (private communication). The method is based on harmonic analysis techniques in the spirit of Chang (Ann. Math. 157 (2003) 939–957) and combinatorics on graphs. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Equidistribution of signs for modular eigenforms of half integral weight

Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato–Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp ²)} _p , where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn ²)} _n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato–Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind–Dirichlet density.     

Infinitely many Gerretsen-Blundon style quadratic inequalities, all strongest in Blundon’s sense

Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s ² ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms q _B (R, r) = 16Rr ? 5r ² and Q _B (R, r) = 4R ² + 4Rr + 3r ²; strongest in the sense that if Q is a quadratic form and s ² ≤ Q(R, r) ≤ Q _B (R, r) for all triangles then Q(R, r) = Q _B (R, r), and similarly for q _B (R, r). In this paper we prove that Q _B (resp. q _B ) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s ² ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.     

Notes on the Kernels of Locally Finite Higher Derivations in Polynomial Rings

Let A = k^[3] be the polynomial ring in three variables over a field k, and let D be a nontrivial locally finite iterative higher derivation on A. Let A^D denote the kernel of D. In this note, we prove that, if chark > 0 and ML(A^D) ≠ A^D, then A^D ? k^[2]. As a consequence of this result, we give another proof of the cancellation theorem for k^[2] over any field k of positive characteristic.     

On the lifting of elliptic cusp forms to cusp forms on quaternionic unitary groups

Let H be a definite quaternion algebra over Q with discriminant DH and R a maximal order of H. We denote by Gn a quaternionic unitary group and put Γn=Gn(Q)∩GL2n(R). Let Sκ(Γn) be the space of cusp forms of weight κ with respect to Γn on the quaternion half-space of degree n. We construct a lifting from primitive forms in Sk(SL₂(Z)) to Sk+2n−2(Γn) and a lifting from primitive forms in Sk(Γ₀(d)) to Sk+2(Γ₂), where d is a factor of DH. These liftings are generalizations of the Maass lifting investigated by Krieg.     

<Emphasis Type="Italic">p</Emphasis>-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight

Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL ₂(ℤ) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ₀(4N) for N = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications to our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.     

Irregular discrepancy behavior of lacunary series

In 1975 Philipp showed that for any increasing sequence (n _k ) of positive integers satisfying the Hadamard gap condition n _k+1/n _k  > q > 1, k ≥ 1, the discrepancy D _N of (n _k x) mod 1 satisfies the law of the iterated logarithm $$ 1/4 \leq {\mathop {\rm lim\,sup} \limits _{N\to\infty}}\, N D_N(n_k x) (N \log \log N)^{-1/2}\leq C_q\quad \textup{a.e.}$$ Recently, Fukuyama computed the value of the lim sup for sequences of the form n _k = θ ^k , θ > 1, and in a preceding paper the author gave a Diophantine condition on (n _k ) for the value of the limsup to be equal to 1/2, the value obtained in the case of i.i.d. sequences. In this paper we utilize this number-theoretic connection to construct a lacunary sequence (n _k ) for which the lim sup in the LIL for the star-discrepancy ${D_N^*}$ is not a constant a.e. and is not equal to the lim sup in the LIL for the discrepancy D _N .     

Spectral asymptotic behavior of elliptic systems

One investigates a first-order elliptic self-adjoint pseudodifferential operator A (x,D) acting in sections of a Hermitian vector bundle over a compactn-dimensional manifold x. It is assumed that the principal symbol A(x, ξ) of the operator is locally diagonalizable and that its eigenvaluesaj(x, ξ) have a variable multiplicity and that {a _i,a _k}≠0 whenevera _i=a _k. Under indicated conditions one constructs an expansion of the fundamental solution of the hyperbolic system \(i\frac{{\partial u}}{{\partial t}} = A(x,D)u\) and one investigates the asymptotic properties of the spectrum of the operator A (x,D). For the distribution functionN(λ) of the eigenvalues one establishes that . Under further assumptions on the properties of the bicharacteristic of the symbolsaj(x, ξ) one establishes a stronger estimate of Duistermaat-Guillemin type:N(λ)=Cλ ⁿ +C′λ ^n?i +0(γ ^n?1 )