I'm very keen to deepen my understanding of arithmetic and diophantine problems. In the past I studied some algebraic, analytic and sieve based number theory. Recently I've been reading Weil – Basic Number Theory which covers some early results of Fermat and Euler in their original forms and then discusses modern interpretations of the work in terms of curves and genus.
Researching into this I think that the next thing I should do is learn arithmetic algebraic geometry: I have been looking for guides and this one was helpful Road map for learning arithmetic algebraic geometry. (Open to other things if they would be more appropriate too).
I am very keen on learning concrete results about the integers, so I have been finding it hard to do things like work through all of Hartshorne before I can learn more number theory. I'm trying to study An Invitation to Arithmetic Geometry – Dino Lorenzini instead, but it seems like it doesn't cover much concrete results either, instead being the background to start studying something like Falting's theorem.
So my question is please can you help recommend to me some results in number theory that I can set as goals to understand the proofs of, and motivate me to follow what looks like a very hard and long path of learning arithmetic geometry. If you could suggest any research papers or books.
I'm having a difficult getting back into mathematics a couple of years after finishing my masters. Since I'm not at an institution I don't have anyone to look to for guidance so I would greatly appreciate help.
Thank you kindly.
Best Answer
That will depend on your more specific interests. One point of entrance into arithmetic geometry could be to have a good understanding of Szpiro's discriminant-conductor inequality for non-isotrivial elliptic curves over a function field, particularly the proof embodied by the Kodaira-Spencer class, and to see how this result (but not its proof!) leads, by direct translation over arithmetic surfaces, to the statement of the $abc$ conjecture (in the averaged form). You may start with Szpiro's papers, particularly Discriminant et conducteur des courbes elliptiques from 1990. This requires a modest algebro-geometric background and will motivate a deeper study of the subject, leading up to works on Arakelov theory and to the Vojta conjectures, which have informed Vojta's diophantine approximations reproof of Faltings's theorem.
Some landmark results in this spirit:
Faltings's Big theorem: For $A/\mathbb{C}$ an abelian variety and $\Gamma \subset A(\mathbb{C})$ a finitely generated subgroup, the Zariski closure of any subset of $\Gamma$ is a finite union of cosets of algebraic subgroups of $A$.
The Manin-Mumford conjecture, proved by Raynaud: The same for $\Gamma = A_{\mathrm{tors}}$, the torsion subgroup of $A$. Szpiro, Ullmo, and Zhang have discovered an Arakelov theoretic proof of a stronger form (Bogomolov's conjecture) that has the torsion points replaced by a sequence of points with canonical height approaching zero.
Andre-Oort conjecture (now a theorem in this case, after papers by Tsimerman and Yuan-Zhang): The description as special subvarieties of Zariski closures of sets of CM points of $\mathcal{A}_g(\mathbb{C})$. This is in the spirit of the other two problems but has especially interesting ties with analytic number theory, which you will easily appreciate given your background.
A guide to current developments on this topic is Zannier's book Some Problems of Unlikely Intersections in Arithmetic and Geometry. And, as for setting a systematic and rigorous learning path, it seems to me that the book you might be looking for is Heights in Diophantine Geometry, by Enrico Bombieri and Walter Gubler. Following it, you could set the proof of Faltings's theorem as a more distant goal while reaping many small rewards continually along the way. The fourth and fifth chapters, for example, prove real results about the integers that do not require any algebro-geomeric background, and yet have extensions leading quickly to arithmetic geometry. Such are the uniform finiteness of solutions to the two-variable $S$-unit equation or the Bogomolov property of the field $\mathbb{Q}^{\mathrm{ab}}$, with a beautiful application to a reproof of Smyth's theorem: an integer non-reciprocal polynomial has Mahler measure bounded away from $1$. All proofs are given in full detail, and all the involved algebro-geometric background is collected in an appendix, that you can learn as you go through the book. After the diophantine approximations proof of Faltings's theorem you may turn with fresh motivation to the specialized literature and engage, for example, with a study of either of the three topics I mentioned earlier.
I have only touched on one facet of arithmetic geometry. But I think it is one that will give you a good taste for the subject. It is then only natural to look into Faltings's Endlickeitsatze paper (that contains among other things the first, and very different, proof of the Mordell conjecture), leading up to other problems and results that, in turn, stimulated developments in arithmetic geometry. In this particular case we have
another major result of arithmetic geometry that is especially well motivated from an analytic number theory background (cf. the Odlyzko bound and Theorem 5.51 in Iwaniec and Kowalski's book).