Linear Combination Of Theta Series Research Articles

Automorphic forms for some even unimodular lattices

We look at genera of even unimodular lattices of rank 12 over the ring of integers of \({{\mathbb {Q}}}(\sqrt{5})\) and of rank 8 over the ring of integers of \({{\mathbb {Q}}}(\sqrt{3})\), using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over \({{\mathbb {Z}}}\), we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift.

Shimura correspondence for level p2 and the central values of L-series, II

Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross–Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.

On Hermitian modular forms of small weight over imaginary quadratic fields

In this paper, we prove that an Hermitian modular form with small weight over the quadratic field with class number one is a linear combination of theta series associated with Hermitian quadratic forms.

Modular forms of weight 8 for Γ g (1, 2)

We complete the program indicated by the Ansatz of D’Hoker and Phong in genus 4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Ansatz. We prove dim[Γ4(1, 2), 8]0 = 2 and dim[Γ4(1, 2), 8] = 7. In each genus, we classify the linear relations among the self-dual lattices of rank 16. We extend the program to genus 5 by constructing the unique linear combination of theta series that satisfies the Ansatz.

Local theta correspondence and the lifting of Duke, Imamoglu and Ikeda

We use results on the local theta correspondence to prove that for large degrees the Duke-Imamoglu-Ikeda lifting of an elliptic modular form is not a linear combination of theta series.

Gitter und Anwendungen

Respecting the different backgrounds of the participants, most of the talks were aimed to a broad audience, among those nine survey talks invited by the organizers. This concept was very successful and opened many discussions and interdisciplinary interactions. It was also very fruitful for the many young participants. Also most of the non speakers took the opportunity to present their recent work via posters. The theory of lattices has many applications and interactions with various other mathematical and technical disciplines such as information technology, topology, algebraic geometry, representation theory, combinatorics, number theory and modular forms to name only the most prominent ones. One of the most classical problems is the construction of dense lattice sphere packings. The densest lattice sphere packings are only known in dimensions up to 8 and, due to a recent work by H. Cohn and A. Kumar, also in dimension 24. A. Kumar gave a very illuminating talk on the strategy of their proof of this great result which uses certain linear programming techniques based on the fact that the minimal vectors of the Leech lattice form a very good design. Design techniques have also been used by B. Venkov to define the notion of strongly perfect lattices, which realize certain local maxima of the density function and which are now classified up to dimension 12. The theory of modular forms is one well established tool in the investigation of lattices as shown in the nice introduction by Krieg. Bannai applies this theory to bound the strength of designs provided by layers of unimodular lattices. On the other hand, lattices provide one important tool for the construction of modular forms. The last talk by B\\"ocherer showed that in many situations all modular forms are linear combinations of theta-series. Of increasing interest but computationally very difficult is the investigation of thin lattice sphere coverings. Vallentin and Sch\\"urmann developed new algorithms to find local optimal coverings and discovered lattice coverings better than the ones previously known. Nguyen presented new reduction algorithms for lattices in small dimensions motivated by practical applications to cryptosystems. For certain applications in information technology not only the density of the lattice but also other properties that minimize the fading error play a role. It turns out that lattices constructed from algebraic number fields yield good codes for Rayleigh fading channels. These ideal lattices are also used to investigate the ring of integers in algebraic number fields as shown in the talk by Schoof and also in the applications of ideal lattices introduced by Bayer-Fluckiger. Strongly related to this is the application to Arakelov theory as illustrated in the talks by Bost and K\\"unnemann. Also Voronoi's classical theory of perfect forms (see Martinet's talk for an introduction) has new fruitful applications in number theory more precisely in the calculation of the homology of $GL_n({\\bf Z})$ as presented by Elbaz-Vincent. Last but not least one should mention the talk by J.-P. Serre on BL-bases and unitary groups in characteristic 2.

Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.

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.

Siegel cusp forms of degree 12k and weight 12k

In this paper we prove the existence of cusp forms relative to the full modular group whose genus is equal to the weight. These cusp forms are linear combination of theta series.

The Eichler Commutation Relation for theta series with spherical harmonics

It is well known that classical theta series which are attached to positive definite rational quadratic forms yield elliptic modular forms, and linear combinations of theta series attached to lattices in a fixed genus can yield both cusp forms and Eisenst

Lattices and θ-function identities. II: Theta series

The gluing construction of lattices is used to generate and study a number of theta function densities. It is shown, for example, that Riemann’s formula, a fundamental degree 4 identity, can be derived from a degree 2 identity. It is proved that all in a large class of identities can be derived from these lattice techniques. A list of 24 independent quadratic identities in the Jacobi functions ϑ1 and ϑ3, conjectured to be complete, with all but two of them seeming to be new, is given. The theta series of glue classes is also investigated, and it is shown that all of them, as well as the functions ϑa, b, satisfy polynomials whose coefficients are linear combinations of theta series of lattices; these polynomials have some interesting properties.

The Dirichlet series of Koecher and Maaß and modular forms of weight 3/2

The basis problem for modular forms is the question which modular forms can be expressed as linear combinations of theta series, in order that this question makes sense one has of course to specify which type of theta series should be taken into account. We want here to follow (in a special case) the approach taken e.g. in [B61, W a l l which (in its general form) asks" Which (cuspidal) modular forms F of degree n and weight k for the group FoC")(N) can be expressed as linear combinations of the theta series of degree n of integral quadratic forms of rank m = 2 k and the same level N? Moreover, one wants to derive an explicit expression for such a linear combination in terms of the Petersson inner products of F with the theta series involved. The representation theoretic version of this question is to ask: Which (cuspidat) automorphic representations of the adelic symplectic (or metaplectic) group

SPHERICAL THETA SERIES

In this paper explicit formulas are obtained which express the images under the action of the Hecke operators of theta series with spherical polynomials and rational translations which correspond to integral quadratic forms in an even number of variables, in terms of linear combinations of theta series of the same type. Transformation formulas are also obtained for such theta series under the action of the integral symplectic group.Bibliography: 7 titles.

THE ACTION OF HECKE OPERATORS ON NONHOMOGENEOUS THETA SERIES

Explicit formulas are obtained which express the images of nonhomogeneous theta series of arbitrary integral nondegenerate quadratic forms in an even number of variables under the action of Hecke operators as linear combinations of theta series of the same type.Bibliography: 4 titles.

Action of the irregular Hecke operator of index p on the theta-series of a quadratic form

Let F be a positive-definite integral even matrix of even order. For an arbitrary prime number p and natural n one obtains an explicit expression for the image of the theta series of genus n of the matrix F, under the action of an irregular Hecke operator with index p, in the form of a linear combination of theta series.

Theta series of weight [formula omitted

We show that a cusp form of weight 3 2 , which corresponds to an Eisenstein series of weight 2 via the Shimura lifting, must be linear combination of theta series attached to quadratic forms of one variable.