Automating the modular method for Q-curves to solve Diophantine equations

This dissertation discusses the modular method for Frey $ \mathbb{Q} $-curves and applies this method to a few new Diophantine problems. In particular it focuses on how certain steps in the modular method can be automated, such that the entire process can easily be applied to a Diophantine equation. This includes both the theory to automate this process as well as an actual implementation as a Python package for SageMath. An important step in applying the modular method is the computation of the conductor of a Frey curve. A careful look at Tate's algorithm shows that the algorithm can also be applied to Frey curves, but in that case each step of the algorithm might have two distinct results based on the value of the parameters. By applying Hensel lifting and keeping track of the cases -- in what we call $\mathfrak{p}$-adic trees -- Chapter 1 demonstrates that Tate's algorithm can be automated for Frey curves to compute the conductor exponent at a prime. The modularity of $ \mathbb{Q} $-curves relates a classical newform to a $ \mathbb{Q} $-curve through an abelian variety of $ \text{GL}_2 $-type. Chapter 2 discusses the theory behind this and explains how to compute the level and character of these newforms from this theory and the conductor computation from Chapter 1. This requires the computation of splitting maps, splitting characters, and the degree map, as well as a potential twist of the original curve, which can all be computed one from the other with some input data about isogenies of the $ \mathbb{Q} $-curve. The chapter also establishes Galois representations associated with $ \mathbb{Q} $-curves and shows how to compute the traces of Frobenius elements under these representations. By applying this theory and level lowering results one can compute newforms associated with Frey $ \mathbb{Q} $-curves of which the level does not depend on the particular value of the parameters anymore. Chapter 3 outlines this procedure and shows a few tactics to then eliminate newforms based on comparing their Galois representation with the corresponding Galois representation of the Frey $ \mathbb{Q} $-curve. Chapter 4 shows that the Diophantine equation $ (x - y)^4 + x^4 + (x + y)^4 = z^n $ has no integer solutions $ (x, y, z, n) $ with $ \gcd(x, y) = 1 $ and $ n > 1 $. It is shown that a particular Hilbert modular approach seems unfeasible for this equation, but that the automated $ \mathbb{Q} $-curve approach works on the same Frey curves to show no solutions exist when $ n > 5 $ is prime. The cases where $ n = 2, 3, 5 $ are shown separately. Chapter 5 considers perfect powers in elliptic divisibility sequences generated by $ \mathbb{Q} $-points on an elliptic curve of $ j $-invariant $ 1728 $. The automated approach is applied to various such examples to prove the non-existence of $ l $-th powers with $ l > l_0 $ a prime number, and $ l_0 $ dependent on the sequence.