On the image of code polynomials under theta map

TL;DR

This paper investigates the properties of the theta map from code polynomials to Siegel modular forms, proving non-surjectivity for genus 4 and discussing embeddings for higher genera.

Contribution

It establishes the non-existence of an embedding for genus 4 and analyzes the surjectivity of the theta map for different genera.

Findings

The theta map is not surjective for g ≥ 4.

No embedding exists between the associated projective varieties for g ≥ 4.

The graded rings are not surjective for g=4.

Abstract

The theta map sends code polynomials into the ring of Siegel modular forms of even weights. Explicit description of the image is known for $g\leq 3$ and the surjectivity of the theta map follows. Instead it is known that this map is not surjective for $g\geq 5$. In this paper we discuss the possibility of an embedding between the associated projective varieties. We prove that this is not possible for $g\geq 4$ and consequently we get the non surjectivity of the graded rings for the remaining case $g=4$.