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# Mathematical Research Letters

## Volume 23 (2016)

### Number 4

### Modular local polynomials

Pages: 973 – 987

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n4.a2

#### Authors

#### 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 developed in this paper to establish an algorithm which explicitly determines the space of modular local polynomials for each $D$.

#### Keywords

local polynomials, modular forms, locally harmonic Maass forms

#### 2010 Mathematics Subject Classification

11E16, 11F11, 11F37

Published 16 September 2016