Explicit Higher Arithmetic Geometry

The PI's research is mainly concerned with Diophantine equations: a Diophantine equation is an equation for which we seek solutions in integers (whole numbers) or rationals (fractional numbers). An example of a Diophantine equation is x^n+y^n=z^n. Fermat's Last Theorem---posed by Fermat 350 years ago and only proved by Wiles in 1995---states that there are no solutions with n at least 3 and x,y,z all non-zero integers. The proof of Fermat's Last Theorem works by relating hypothetical solutions of the Fermat equation to elliptic modular forms via a Frey elliptic curve. In the work of Jarvis (Sheffield) and of Darmon (McGill) a generalization of this setting is envisaged where solutions of Diophantine equations are related to Hilbert modular forms via Frey elliptic curves over number fields or via Frey hypergeometric Abelian varieties. It is proposed to investigate this approach and make it explicit for several families of Diophantine equations, which may then be solved with the help of recent computational breakthroughs due to Dembele.Another direction of the proposed study involves the explicit arithmetic of subvarieties of Abelian varieties. Such varieties are the subject of recent theoretical advances by Faltings, Vojta, Buium, etc. In many ways, these varieties are the most natural generalization of curves of higher genus who explicit arithmetic has been intensively studied by Cassels, Flynn, Schaefer, Poonen, Stoll, Bruin, etc. over the last 15 years. The proposed research will seek to transfer many of the techniques applicable to curves to the realm of subvarieties of Abelian varieties. In particular, we will seek analogues of Coleman bounds, Chabauty, Mordell-Weil and explicit methods for determining rational points.