I would like to learn new techniques for solving diophantine equations. I know how to solve diophantine equations with factorization over $\mathbb Z$ and $\mathbb Z[i]$, modular arithmetic, the Liftng The Exponent lemma and other elementary techniques, but I would like to see some more advanced techniques.
To be particular, I am interested in solving Diophantine equations by using results from Algebraic Number Theory and Algebraic Geometry.
I had a few lessons in Algebraic Number Theory and Alebraic Geometry. What topics from each of the pre-mentioned subjects would you recommend to emphasize on, if one wants to strengthen his/her Diophantine equation-solving skills?
I would also appreciate any book recommendations.
Thanks in advance!