Let M (n) k be the space of Siegel modular forms of degree n and even weight k. In this paper firstly a certain subspace Spez(M (2n) k ) the Spezialschar of M (2n) k is introduced. In the setting of the Siegel three-fold it is proven that this Spezialschar is the Maass Spezialschar. Secondly an embedding of M (2) k into a direct sum ⊕ ⌊ k 10 ⌋ ν=0 Sym 2 Mk+2ν is given. This leads to a basic cha...

A moduli space is a space that parametrizes geometric objects. For example, elliptic curves are classified by the so-called J-invariant, so the moduli space of elliptic curves is a line (with coordinate J). More generally, there exists a moduli space, calledMg , which parametries all projective algebraic curves of genus g (equivalently, all compact Riemann surfaces of genus g). The Jacobian of ...

The moduli space of quadratic differentials is not compact: when some loop on a Riemann surface M becomes short in the flat metric defined by a quadratic differential q, the corresponding point (M, q) of the moduli space tends to a “cusp”. We describe typical degenerations, thus describing “generic cusps of moduli space”. The part of the boundary of the moduli space which does not arise from “g...

In this article, we exposit a proof, due to Kollár in [9], of the fact that the moduli stack Mg of stable curves of genus g ≥ 2 admits a coarse moduli space Mg which is projective over Z. In particular, this means that the coarse moduli space Mg, which is a priori but an algebraic space, is actually a projective scheme over Z. Together with the work of Deligne–Mumford [3], the moduli space Mg i...

For two positive integers m and n, we let Hn be the Siegel upper half plane of degree n and let C be the set of all m × n complex matrices. In this article, we investigate differential operators on the Siegel-Jacobi space Hn ×C(m,n) that are invariant under the natural action of the Jacobi group Sp(n,R)⋉ H (n,m) R on Hn × C, where H R denotes the Heisenberg group.

A result of Chai–Faltings on Satake parameters of Siegel cusp forms together with the classification of unitary, unramified, irreducible, admissible representations of GSp4 over a p-adic field, imply that the local components of the automorphic representation of GSp4 attached to a cuspidal Siegel eigenform of degree 2 must lie in certain families. Applications include estimates on Hecke eigenva...

Using a birational correspondence between the twistor space of S2n and projective space, we describe, up to birational equivalence, themoduli space of superminimal surfaces in S2n of degree d as curves of degree d in projective space satisfying a certain differential system. Using this approach, we show that the moduli space of linearly full maps is nonempty for sufficiently large degree and we...

Abstract—Through the Fukaya conjecture and the wrapped Floer cohomology, the correspondences between paths in a loop space and states of a wrapping space of states in a Hamiltonian space (the ramification of field in this case is the connection to the operator that goes from TM to T*M) are demonstrated where these last states are corresponding to bosonic extensions of a spectrum of the space-ti...