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 haven't read Algebraic Number Theory and I just had a few lessons in Alebraic Geometry in the present semester. Thus, I may need to read a lot.
What topics from each of the pre-mentioned subjects do you resommend me to read and emphasize on, in order to strengthen my Diophantine equation-solving skills?
I would also appreciate any book recommendations.
Thanks in advance!