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

@article{Clry2016CovariantsOB, title={Covariants of binary sextics and vector-valued Siegel modular forms of genus two}, author={Fabien Cl{\'e}ry and Carel Faber and Gerard van der Geer}, journal={Mathematische Annalen}, year={2016}, volume={369}, pages={1649-1669} }

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 degree 2. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree 2.

## 17 Citations

### Covariants of binary sextics and modular forms of degree 2 with character

- Mathematics, Computer ScienceMath. Comput.
- 2019

For a modular form defined by a covariant, the order of vanishing along the locus of products of elliptic curves in terms of the covariant is expressed.

### 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…

### Generating Picard modular forms by means of invariant theory

- Mathematics
- 2021

We use the description of the Picard modular surface for discriminant −3 as a moduli space of curves of genus 3 to generate all vector-valued Picard modular forms from bi-covariants for the action of…

### 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.

### Modular forms via invariant theory

- Mathematics
- 2022

. We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The ﬁrst one deals with elliptic modular forms and their…

### Modular forms of degree two and curves of genus two in characteristic two

- Mathematics, Biology
- 2020

The ring of modular forms of degree 2 in characteristic 2 is described using its relation with curves of genus 2 as a guide to solve the inequality of theorems of EPTs.

### Modular Forms of Degree 2 and Curves of Genus 2 in Characteristic 2

- Mathematics, BiologyInternational Mathematics Research Notices
- 2020

The ring of modular forms of degree two in characteristic two is described using its relation with curves of genus two using the principle of entailment.

### Genus 2 paramodular Eisenstein congruences

- Mathematics
- 2016

We investigate certain Eisenstein congruences, as predicted by Harder, for level p paramodular forms of genus 2. We use algebraic modular forms to generate new evidence for the conjecture. In doing…

### Computing isogenies from modular equations between Jacobians of genus 2 curves

- Mathematics, Computer Science
- 2020

An algorithm is presented solving the following problem: given two genus 2 curves over k with isogenous Jacobians, compute an isogeny between them explicitly, which can be either an {\ell}-isogeny or, in the real multiplication case, an isogene with cyclic kernel.

## References

SHOWING 1-10 OF 28 REFERENCES

### Siegel modular forms of genus 2 and level 2 (with an appendix by Shigeru Mukai)

- Mathematics
- 2015

We study vector-valued Siegel modular forms of genus 2 on the three level 2 groups Γ[2] ◁ Γ1[2] ◁ Γ0[2] ⊂ Sp(4, ℤ). We give generating functions for the dimension of spaces of vector-valued modular…

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

- Mathematics, Computer Science
- 2014

The method is efficient in producing the siblings of Delta, the smallest weight cusp forms that appear in low degrees, and shows the strong relations between these modular forms of different genera.

### 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…

### 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…

### Decomposable Ternary Cubics

- MathematicsExp. Math.
- 2002

Cubic forms in three variables are parametrised by points of a projective space p9 and their equivariant minimal resolutions are calculated and their ideals invariant-theoretically are described.

### ON SIEGEL MODULAR FORMS OF GENUS TWO (II).1

- Mathematics
- 1964

Introduction. The main subject we shall discuss in this second paper is the Siegel modular forms of genus two with levels. The method we used in the first paper [5] did not give sufficient…

### Elliptic modular forms and their applications.

- Mathematics
- 2008

These notes give a brief introduction to a number of topics in the classical theory of modular forms, based on various courses held at the College de France in the years 2000–2004.

### Congruence Between a Siegel and an Elliptic Modular Form

- Mathematics
- 2008

The winter semester 2002/2003 was the last semester before my retirement from the university. It also happened that I was the chairman of the Colloquium and the speaker foreseen for February 7 had to…

### On Invariant Theory

- Mathematics, Computer Science
- 1995

In this paper it is shown how the invariants of n−ary forms can be produced from the discriminant of multilinear forms (determinants of multidimensional matricies), which should be considered as the generalization of the operation of taking classical hessians and resultants.

### A Generalized Jacobi Theta Function and Quasimodular Forms

- Mathematics
- 1995

In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \( {\tilde…