In this article, we establish explicit and uniform L^∞-norm bounds for L^2-normalized Siegel–Jacobi cusp forms of integral weight k and index m for the Siegel modular group Γ_0=Sp_2g(ℤ) for arbitrary genus g≥ 1. Using the generalization of the classical Eichler–Zagier theta decomposition to higher genus, any such Siegel–Jacobi cusp form can be written as a finite linear combination of Siegel cusp forms of half-integral weight k-1/2 multiplied by the higher-dimensional analogues of the classical Jacobi theta functions. By building upon the uniform L^∞-norm bounds on average for Siegel cusp forms established by J. Kramer and A. Mandal via the associated Bergman kernels, we prove that for k∈ℤ_≥ g+1, m∈ℤ_≥ 1, and a given ε>0, the L^∞-norm bound ‖_L^∞=sup_(τ,z)∈ℍ_g×ℂ^g(τ,z)‖_Pet=O_ Γ_0,ε(k^(3g^2+5g)/8 m^g^2+5g/4+ε) holds for any Siegel–Jacobi cusp form ϕ that is L^2-normalized with respect to the Petersson inner product. These estimates provide the first explicit upper bounds in terms of both parameters k and m for arbitrary genus g.

In this article, we give L^∞ -norm bounds for the natural invariant norm of cusp forms of real weight k and character χ for any cofinite Fuchsian subgroup Γ⊂SL_2(ℝ) . Using the representation of Jacobi cusp forms of integral weight k and index m for the modular group Γ _0=SL_2(ℤ) as linear combinations of modular forms of weight k-1/2 for some congruence subgroup of Γ _0 (depending on m) and suitable Jacobi theta functions, we derive L^∞ -norm bounds for the natural invariant norm of these Jacobi cusp forms. More specifically, letting J_k,m^cusp(Γ _0) denote the complex vector space of Jacobi cusp forms under consideration and ‖·‖ _Pet the pointwise Petersson norm on J_k,m^cusp(Γ _ 0) , we prove that for k∈ℤ_≥ 5 and m∈ℤ_≥ 1 , and a given ϵ >0 , the L^∞ -norm bound ‖ϕ‖ _L^∞=sup _(τ ,z)∈ℍ×ℂ‖ϕ (τ ,z)‖ _Pet=O_Γ _0,ϵ (k m^7/4+ϵ ) holds for any ϕ∈ J_k,m^cusp(Γ _0) , which is L^2 -normalized with respect to the Petersson inner product, where the implied constant depends on Γ _0 and the choice of ϵ >0 .

In this paper we extend the arithmetic intersection theory of adelic divisors on quasiprojective varieties developed by X. Yuan and S. W. Zhang to cover certain adelic arithmetic divisors that are not nef nor integrable. The key concept used in this extension is the relative finite energy introduced by T. Darvas, E. Di Nezza, and C. H. Lu. As an application we compute the arithmetic self-intersection number of the line bundle of Siegel-Jacobi forms on the universal abelian variety endowed with its invariant hermitian metric. The techniques developed in this paper can be applied in many other situations like mixed Shimura varieties or the moduli space of stable marked curves.

Zusammenfassung In diesem Beitrag werden die bahnbrechenden Ergebnisse von Maryna Viazovska zu Kugelpackungen in höheren Dimensionen vorgestellt.

In this paper we investigate some number theoretic properties of the frequencies of the Korteweg-de Vries equation on the torus, relevant for the stability of finite gap solutions.

In diesem Beitrag soll über die neuesten, aufsehenerregenden Entwicklungen im Zusammenhang mit der Vermutung von Fermat berichtet werden. Diese Vermutung besagt, dass es keine von Null verschiedenen, ganzen Zahlen a, b, c gibt, welche der Gleichung 1 $$ a^n + b^n = c^n $$ genügen, sobald der Exponent n größer als zwei ist. Fermat stellte seine Vermutung um das Jahr 1637 herum, also vor mehr als 350 Jahren, auf.

Let Γ⊊Sp_n(ℝ) be an arithmetic subgroup of the symplectic group Sp_n(ℝ) acting on the Siegel upper half-space ℍ_n of degree n. Consider the d-dimensional space of Siegel cusp forms 𝒮_κ^n(Γ) of weight κ for Γ and let {f_j}_1≤ j≤ d be a basis of 𝒮_κ^n(Γ) orthonormal with respect to the Petersson inner product. In this paper we show using the heat kernel method that the sup-norm of the quantity S_κ^Γ(Z):=∑_j=1^d (Y)^κ|f_j(Z)|^2 (Z∈ℍ_n) is bounded above by c_n,Γκ^n(n+1)/2 when M:=Γ\ℍ_n is compact and by c_n,Γκ^3n(n+1)/4 when M is non-compact of finite volume, where c_n,Γ denotes a positive real constant depending only on the degree n and the group Γ. Furthermore, we show that this bound is uniform in the sense that if we fix a group Γ_0 and take Γ to be a subgroup of Γ_0 of finite index, then the constant c_n,Γ in these bounds depends only on the degree n and the fixed group Γ_0.

In this paper we generalize a well-known isomorphism between the space of cusp forms of weight k for a Fuchsian subgroup of the first kind $$\Gamma \subset \mathrm {SL}_{2}({\mathbb {R}})$$ and the space of certain Maaß forms of weight k for $$\Gamma $$ to an isomorphism between the space of Siegel cusp forms of weight k for a subgroup $$\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})$$ , which is commensurable with the Siegel modular group $$\mathrm {Sp}_{n}({\mathbb {Z}})$$ , and a suitable space of Siegel–Maaß forms of weight k for $$\Gamma $$ .

Let Γ ⊂ P S L 2 ( R ) \Gamma \subset \mathrm {PSL}_{2}(\mathbb {R}) be a Fuchsian subgroup of the first kind acting on the upper half-plane H \mathbb {H} . Consider the d 2 k d_{2k} -dimensional space of cusp forms S 2 k Γ \mathcal {S}_{2k}^{\Gamma } of weight 2 k 2k for Γ \Gamma , and let { f 1 , … , f d 2 k } \{f_{1},\ldots ,f_{d_{2k}}\} be an orthonormal basis of S 2 k Γ \mathcal {S}_{2k}^{\Gamma } with respect to the Petersson inner product. In this paper, we will give effective upper and lower bounds for the supremum of the quantity S 2 k Γ ( z ) := ∑ j = 1 d 2 k | f j ( z ) | 2 I m ( z ) 2 k S_{2k}^{\Gamma }(z):=\sum _{j=1}^{d_{2k}}\vert f_{j}(z)\vert ^{2}\,\mathrm {Im}(z)^{2k} as z z ranges through H \mathbb {H} .