# Constructing vector-valued Siegel modular forms from scalar-valued Siegel modular forms

@article{Clery2014ConstructingVS, title={Constructing vector-valued Siegel modular forms from scalar-valued Siegel modular forms}, author={Fabien Cl'ery and Gerard van der Geer}, journal={arXiv: Algebraic Geometry}, year={2014} }

This paper gives a simple method for constructing vector-valued Siegel modular forms from scalar-valued ones. The method is efficient in producing the siblings of Delta, the smallest weight cusp forms that appear in low degrees. It also shows the strong relations between these modular forms of different genera. We illustrate this by a number of examples.

## 11 Citations

### On vector-valued Siegel modular forms of degree 2 and weight (j,2)

- Mathematics
- 2017

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of…

### Covariants of binary sextics and vector-valued Siegel modular forms of genus two

- Mathematics
- 2016

We extend Igusa’s description of the relation between invariants of binary sextics and Siegel modular forms of degree 2 to a relation between covariants and vector-valued Siegel modular forms of…

### Heat equation for theta functions and vector-valued modular forms

- Mathematics
- 2015

We give a method for constructing vector-valued modular forms from singular scalar-valued ones of a suitable type. As an application, we prove that two remarkable spaces of vector-valued modular…

### Minimal modularity lifting for nonregular symplectic representations

- Mathematics
- 2019

In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series…

### Exploring modular forms and the cohomology of local systems on moduli spaces by counting points

- Mathematics
- 2016

This is a report on a joint project in experimental mathematics with Jonas Bergstr\"om and Carel Faber where we obtain information about modular forms by counting curves over finite fields.

### Siegel modular forms of degree two and three and invariant theory

- Mathematics
- 2021

. This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Cl´ery and Carel Faber.

### Automorphic Forms on Unitary Groups

- Mathematics
- 2022

This manuscript provides a more detailed treatment of the material from my lecture series at the 2022 Arizona Winter School on Automorphic Forms Beyond GL2. The main focus of this manuscript is…

### Analytic evaluation of Hecke eigenvalues for Siegel modular forms of degree two

- MathematicsThe Open Book Series
- 2019

The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform F is to determine a large number of Fourier coefficients of F and then compute the Hecke action on those…

### Concomitants of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three

- MathematicsSelecta Mathematica
- 2020

We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of…

### Formes automorphes et voisins de Kneser des réseaux de Niemeier

- Mathematics
- 2014

In this memoir, we study the even unimodular lattices of rank at most 24, as well as a related collection of automorphic forms of the orthogonal and symplectic groups of small rank. Our guide is the…

## References

SHOWING 1-10 OF 22 REFERENCES

### Siegel Modular Forms and Their Applications

- Mathematics
- 2008

These are the lecture notes of the lectures on Siegel modular forms at the Nordfjordeid Summer School on Modular Forms and their Applications. We give a survey of Siegel modular forms and explain the…

### Siegel modular forms of degree three and the cohomology of local systems

- Mathematics
- 2011

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space $$\mathcal{A }_3$$ of principally polarized abelian…

### Level one algebraic cusp forms of classical groups of small ranks

- Mathematics
- 2012

We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over Q of any given infinitesimal character, for essentially all n <= 8. For this, we…

### A Siegel cusp form of degree 12 and weight 12

- Mathematics, Computer Science
- 1998

The theta series of the two unimodular even positive definite lattices of rank 16 are known to be linearly dependent in degree at most 3 and linearly independent in degree 4 and the resulting Siegel cusp form of degree 12 and weight 12 is a Hecke eigenform which seems to have interesting properties.

### EULER PRODUCTS CORRESPONDING TO SIEGEL MODULAR FORMS OF GENUS 2

- Mathematics
- 1974

In this article we construct a theory of Dirichlet series with Euler product expansions corresponding to analytic automorphic forms for the integral symplectic group in genus 2; in Chapter 2 we…

### Schottky’s Invariant and Quadratic Forms

- Mathematics
- 1981

In his classical paper on the moduli of 4 dimensional principally polarized abelian varieties Schottky introduced a homogeneous polynomial J of degree 16 in the Thetanullwerte which vanishes at every…

### Modular forms vanishing at the reducible points of the Siegel upper-half space.

- Mathematics
- 1983

The outline of the paper is s follows: In Sect. l we recall the addition and the transformation formulas of theta functions, which are studied in [3], [4]. We shall in Sect. 2 give a criterion (Th.…

### On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n

- Mathematics
- 2001

i=1 (cf. [8]). In this paper we will prove that their conjecture is true. Moreover, we obtain a simple Fourier coefficient formula for F(Z). Ibukiyama [23] also made a similar conjecture which was…

### Eigenvalues of Ikeda lifts

- Mathematics
- 2015

In this paper, we compute explicit formulas for the Hecke eigenvalues of Ikeda lifts. These formulas, though complicated, are obtained by purely elementary techniques.

### Computations of Spaces of Siegel Modular Cusp Forms (保型形式およびそれに付随するディリクレ級数の研究 研究集会報告集)

- Mathematics, Materials Science
- 2002

We survey the known dimensions of $s_{n}^{k}$ , the space of Siegel modular forms of weight it and degree $n$ . We mention afew new results for degrees 4, 5and 6. We obtain our results by combining…