Jacobi Maaß forms

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J _k,m ^nh ) of weight k∈? and index m∈? as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\). We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J _k,m ^nh . We construct new examples of cuspidal Jacobi Maaß forms F _f of weight k∈2? and index 1 from weight k?1/2 Maaß forms f with respect to Γ₀(4) and show that the map f ? F _f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J _k,m ^nh can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.     

Ando’s theorem for nonnegative forms

In this paper, we present a generalization of Ando??s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et?al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1?C12, 1999). That is, [A]B??? [B]A or [B]A??? [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form ${\mathfrak{t}}$ into an almost dominated and a singular part with respect to a nonnegative form ${\mathfrak{w}}$ (see Hassi et?al. in J. Funct. Anal. 257(12), 3858?C3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms.     

Kato–Milne cohomology and polynomial forms

Given a prime number p, a field F with $\mathrm{char}(F)=p$ and a positive integer n, we study the class-preserving modifications of Kato–Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and p-regular forms. A p-regular form is defined to be a homogeneous polynomial form of degree p for which there is no nonzero point where all the order $p?1$ partial derivatives vanish simultaneously. We define a ${\stackrel{?}{C}}_{p,m}$ field to be a field over which every p-regular form of dimension greater than $p^m$ is isotropic. The main results are that for a ${\stackrel{?}{C}}_{p,m}$ field F, the symbol length of $H_p^2(F)$ is bounded from above by $p^{m?1}?1$ and for any $n??(m?1){\log }_2?(p)?+1$, $H_p^{n+1}(F)=0$.     

Normal forms under Simon’s congruence

Simon’s congruence, denoted by \(\sim _k\), relates the words having the same subwords of length at most k. In this paper a normal form is presented for the equivalence classes of \(\sim _k\). The length of this normal form is the shortest possible. Moreover, a canonical solution of the equation \(pwq\sim _k r\) is also shown (the words p, q, r are parameters), which can be viewed as a generalization of giving a normal form for \(\sim _k\). In this paper, there can be found an algorithm with which the canonical solution can be determined in \(O((L+n)n^{k})\) time, where L denotes the length of the word pqr and n is the size of the alphabet.     

Stiefel–Whitney invariants for bilinear forms

We examine potential extensions of the Stiefel–Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt?s Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)     

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.     

Sharp forms of nevanlinna’s error terms

Let f(z) be a meromorphic function in the plane. If ψ(t)/t andp(t) are two positive, continuous and non-decreasing functions on [1,∞) with ∫ ₁ ^∞ dt/ψ(t) = ∞ and ∫ ₁ ^∞ dt/p(t) = ∞, then asr → ∞ outside a small exceptional set, provided that the divergence of the integral ∫ ₁ ^r dt/ψ(t) is slow enough. The same forms for the logarithmic derivative and for the ramification term are obtained. It is shown by example that the estimates are best possible. Author supported by Max-Planck-Gesellschaft Z.F.D.W and by NSFC.     

Berezin–Toeplitz quantization for lower energy forms

Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k^?N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.     

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyén, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.     

Generators of Jacobi forms are Poincaré series

The ring of Jacobi forms of even weights is generated by the weak Jacobi forms \(\phi _{-2,1}\) and \(\phi _{0,1}\). Bringmann and the first author expressed \(\phi _{-2,1}\) as a specialization of a Maass–Jacobi–Poincaré series. In this paper, we extend the domain of absolute convergence of Maass–Jacobi–Poincaré series which allows us to show that \(\phi _{0,1}\) is also a Poincaré series.     

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.     

Hardy’s theorem for zeta-functions of quadratic forms

LetQ(u ₁,…,u ₁) =Σd _ij u _i u _j (i,j = 1 tol) be a positive definite quadratic form inl(≥3) variables with integer coefficientsd _ij (=d _ji ). Puts=σ+it and for σ>(l/2) write $$Z_Q (s) = \Sigma '(Q(u_1 ,...,u_l ))^{ - s} ,$$ where the accent indicates that the sum is over alll-tuples of integer (u ₁,…,u _l ) with the exception of (0,…, 0). It is well-known that this series converges for σ>(l/2) and that (s-(l/2))Z _Q (s) can be continued to an entire function ofs. Let σ be any constant with 0<σ<1/100. Then it is proved thatZ _Q (s)has ?_δTlogT zeros in the rectangle(|σ-1/2|≤δ, T≤t≤2T).     

A conjecture on the Eichler cohomology of automorphic forms

We prove partially a conjecture of Knopp about the Eichler cohomology of automorphic forms on H-groups.     

Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians

We hypothesize the form of a transformation reducing the elliptic A _N Calogero–Moser operator to a differential operator with polynomial coefficients. We verify this hypothesis for N ≤ 3 and, moreover, give the corresponding polynomial operators explicitly.     

The Eichler–Shimura cohomology theorem for Jacobi forms

Let \(\Gamma \) be a subgroup of finite index in \(\mathrm {SL}(2,\mathbb {Z})\). Eichler defined the first cohomology group of \(\Gamma \) with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on \(\Gamma \) with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups \(\Gamma ^{(1,j)}\) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if \(j=1\) and the weights of Jacobi cusp forms are in \(\frac{1}{2}\mathbb {Z}-\mathbb {Z}\), then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.     

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.     

Two closed forms for the Apostol–Bernoulli polynomials

In this note, we shall obtain two closed forms for the Apostol–Bernoulli polynomials.