Modular units are functions on modular curves whose divisors are supported only on cusps.
In this talk we’ll discuss how Siegel functions can be used to construct a basis for the group of modular units on $X_{1}\left( p \right)$. One of the main applications of modular units is the study of the cuspidal group  - the subgroup of the Jacobian generated by the cusps. We’ll discuss how our modular units are used in the study of the cuspidal group of $X_{1}\left( p \right)$ and more generally the rational torsion subgroup of the Jacobian.