DERIVED CATEGORY METHODS IN ARITHMETIC: AN APPROACH TO SZPIRO'S CONJECTURE VIA HOMOLOGICAL MIRROR SYMMETRY AND BRIDGELAND STABILITY CONDITIONS

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The arithmetic of elliptic curves occupies a central role in number theory and Diophantine geometry. Diophantine geometry studies Diophantine equations, that is, the solution of polynomial equations in integers or rational numbers (in the most basic case), through a combination of techniques from algebraic geometry, algebraic and analytic number theory, and complex geometry. Szpiro's conjecture for elliptic curves over number fields is known to imply the famous abc-conjecture, whose validity in turn yields a large number of other deep results such as Fermat's Last Theorem, Mordell's Conjecture (Falting's theorem), or Roth's theorem about Diophantine approximation of algebraic numbers. Szpiro's conjecture in the arithmetic set-up has an analogue in complex geometry, relating the number of critical points and the number of singular fibres of a non-trivial semistable family of elliptic curves over some base curve (or more generally, curves of higher genus, due to A. Beauville); Szpiro's inequality also has an analogue in symplectic geometry established by Amoros, Bogomolov, Katzarkov, Pantev, whose proof is essentially a topological/group-theoretic argument involving the mapping class group of a torus with one hole. Homological Mirror Symmetry is a principle/yoga having its origin in mathematical physics, whose consequences mathematicians have only started fully to exploit and understand. In particular, it relates symplectic geometry and complex geometry in completely unexpected ways. For example, graded symplectic automorphisms of a torus can be related to autoequivalences of the derived category of coherent sheaves on the mirror elliptic curve, and Dehn twists are seen to correspond to so-called spherical twists. One can then seek to mimic parts of the proof by Amoros, Bogomolov, Katzarkov, Pantev working with derived autoequivalences and using changes in Bridgeland phase as a substitute for the notion of displacement angle in the symplectic situation. It is reasonable to hope that such an argument will still make sense for arithmetic elliptic fibrations and can lead to a proof of Szpiro's conjecture. The goal of the project is to establish foundations and a framework in which Bridgeland stability conditions can be made sense of in arithmetic/Arakelov geometry and in which the programme inspired by Homological Mirror Symmetry outlined above can be carried through. This will also ultimately involve techniques from p-adic geometry and Berkovich spaces.