Modular local polynomials

TL;DR

This paper studies modular local polynomials, establishing dimension bounds related to geodesics from quadratic forms, and provides an algorithm to explicitly determine their spaces for each discriminant D.

Contribution

It introduces new dimension inequalities for modular local polynomials and develops an explicit algorithm for their determination based on the discriminant D.

Findings

Dimension of space is maximal if and only if D is an even square.

Established an inequality for the dimension of modular local polynomials.

Provided an algorithm to explicitly compute these spaces for any discriminant D.

Abstract

In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when the exceptional set is given by certain natural geodesics related to binary quadratic forms of (positive) discriminant $D$. We furthermore show that the dimension is the largest possible if and only if $D$ is an even square. Following this, we describe how to use the methods developped in this paper to establish an algorithm which explicitly determines the space of modular local polynomials for each $D$.