## STNB2020(34th edition)

### Automating the modular method for Frey Q-curves

Joey van Langen

#### Abstract

The modular method has effectively been applied to solve a variety of exponential Diophantine equations. Although the procedure is in essence very similar for every case, only implementations for particular Diophantine equations can be found in the literature. In this talk I will describe the mathematical obstacles to generalizing such approaches. The talk will focus on a general algorithm that can be applied to problems for which a Frey Q-curve exists. An implementation of this work in Sage provides a simple way to obtain the results of the modular method for such problems. As an illustration we will use this approach on the Diophantine equation $(x - y)^4 + x^4 + (x + y)^4 = z^l$.