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!

  • $\begingroup$ In books usually write the classics. There are new methods not described. Although probably better to say what the equation of interest? $\endgroup$ – individ Mar 20 '17 at 10:15
  • $\begingroup$ swc.math.arizona.edu/aws/2006/06CohenLectures.pdf $\endgroup$ – i9Fn Mar 20 '17 at 10:23

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