Matthew Welsh

A parametrization of roots of polynomial congruences and applications

We discuss a parametrization of roots of polynomial congruences derived using a correspondence between the roots and ideals in orders of number fields. In the quadratic case the parametrization is due to Gauss and has numerous arithmetic applications, often providing an entrance for the spectral theory of SL2 automorphic forms. We also discuss some possibilities and obstructions for higher degree analogues.