Fabien Cléry's research while affiliated with Loughborough University and other places
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Publications (19)
Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level two structure, together with a computation of Euler characteristics we find the isotypical decomposition, under the symmetric group on 6 letters, of spaces of vector-valued Siegel modular forms of degree two and level two.
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 derivatives, and generalizes the Rankin-Cohen bracket, while the second one deals with vector-valued modular forms of genus greater than 1.
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 derivatives, and generalizes the Rankin-Cohen bracket, while the second one deals with vector-valued modular forms of genus greater than one.
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 ${GL}_2$ on the space of pairs of binary forms of bidegree $(4,1)$. The universal binary forms of degree $4$ and $1$ correspond to a meromorphic m...
We describe the ring of modular forms of degree $2$ in characteristic $2$ using its relation with curves of genus $2$.
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 vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays...
Equations of dispersionless Hirota type have been thoroughly investigated in the mathematical physics and differential geometry literature. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional and the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the st...
We describe the ring of modular forms of degree 2 in characteristic 2 using its relation with curves of genus 2.
We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichm\"uller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus play...
We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular form defined by a covariant we express the order of vanishing along the locus of products of elliptic curves in...
We extend Igusa's description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.
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 weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appen...
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 cusps 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 ex...
We construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.
We study vector-valued Siegel modular forms of genus 2 and level 2. We
describe the structure of certain modules of vector-valued modular forms over
rings of scalar-valued modular forms.
We construct generators for modules of vector-valued Picard modular forms on
a unitary group of type (2,1) over the Eisenstein integers. We also calculate
eigenvalues of Hecke operators acting on cusp forms.
In this paper, we study Jacobi forms of half-integral index for any even integral positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A
1=〈2〉). We construct Jacobi forms of singular (respectively, critical) weight in all dimensions n≥8 (respectively, n≥9). We give the Jacobi lifting for...
We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the
paramodular groups of genus 2 vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of
a question formulated during the conference ‘Black holes, Black Rings and Modular Forms’ (ENS, Paris, Au...
Citations
... Similarly, for degree 3 we apply it to the invariant theory of ternary quartics and recover the method of the paper [3] that allows one to construct all Siegel modular forms on Sp(6, Z) from one cusp form χ 4,0,8 of weight (4,0,8). ...
Reference: Modular forms via invariant theory
... One can define a relative notion of a Picard curve for any scheme S defined over Z[1/3, ρ], where ρ = e 2πi/3 ; see [AP07]. The stacks N and N from [CvdG22] can thus be defined more generally over Z[1/3, ρ]. We can define Teichmüller modular forms by pulling back the Hodge bundle E on A 3 under the Torelli map N ct → A 3 . ...
... These could be called Teichmüller motives. Conjecturally, the first two such instances occur for g = 3 and λ = (11, 3, 3) or (7, 7, 3) and correspond to two of the seven automorphic representations of weight 23 in [CT20, Theorem 3], as described in [CFvdG20]. Conversely, for each λ with |λ| = n, there are elements b λ,µ ∈ Z[L] such that ...
... Paper [16] studies integrability of dispersionless Hirota-type equations in 3D, F (u x i x j ) = 0, where u(x 1 , x 2 , x 3 ) is a function of three independent variables, and u x i x j denote second-order partial derivatives. It was shown in [10] that the 'generic' integrable Hirota master-equation is expressible via genus three theta constants. In particular, for equations of the form ...
Reference: On ODEs satisfied by modular forms
... Igusa [14] used this to describe the generators for the rings of scalar-valued Siegel modular forms of degree 2 and later Tsuyumine [29] extended this to the case of degree 3. In joint work with Carel Faber [2,3] we used the description of M 2 as a stack quotient of GL 2 to extend the work of Igusa by describing how invariant theory makes it possible to efficiently generate all vector-valued Siegel modular forms (of level 1) of degree 2 from one universal vector-valued meromorphic Siegel modular form χ 6,−2 and one scalarvalued holomorphic form χ 10 . Similarly in [4] we used the description of an open part of M 3 as a stack quotient of GL 3 to generate all Siegel and Teichmüller modular forms from a universal meromorphic Teichmüller modular form χ 4,0,−1 of genus 3 and the form χ 9 , a square root of a Siegel modular form χ 18 . ...
... For the non-abelian groups G, the information that is missing is an eigenvalue λ γ for all γ ∈ G. This problem is solved for the quaternionic groups Q 4n by noting that it consists of the 11 matrices ±S j and ±U S j for j = 1, . . . , n, and the latter all have eigenvalues ǫ 4 , −ǫ 4 . ...
... We briefly recall the notion of Picard modular forms on the 2-ball. We refer to [9,5] for more details. Let F = Q( √ −3) with ring of integers O F = Z[ρ] for a primitive third root of unity ρ and units O × F = µ 6 . ...
... For example, in the case of = 2 there is the modular form 6,8 , a section of Sym 6 ( ) ⊗ det( ) 8 , that appeared in [4] as follows. Recall that the Torelli morphism 2 ↪ 2 has a dense image and we have an equality of standard compactifications 2 = 2 . ...
... Consider a period matrix τ ∈ H 2 attached to a p.p. abelian surface A over C. Then the period matrices of abelian surfaces linked to A by an isogeny of a given type can be computed by letting certain symplectic matrices act on τ . This precisely corresponds to analytic formulas for the action of Hecke operators on spaces of Siegel modular forms [CG15,§10], [Kri90, Chap. VI, §5]. ...
... In this paper, we refine the previously known dimension formulas for spaces M k, j (Γ [2]) of vectorvalued Siegel modular forms of degree 2 and level 2, by determining their isotypical decomposition under the action of Sp(4, F 2 ) ∼ = S 6 . This extends previous work, see for instance [18,1,28,29,16,17,30,12]. In particular, Tsushima gave in [29,Theorems 2,3] a formula for the dimension of the space S k, j (Γ[N ]) for any N under the conditions j 1 and k 5 or j = 0 and k 4. The ranges for j and k in Tsushima's dimension formula for N = 2 have been slightly extended in [12,Theorem 12.1]. ...
