DERIVED CATEGORY METHODS IN ARITHMETIC: AN APPROACH TO SZPIRO'S CONJECTURE VIA HOMOLOGICAL MIRROR SYMMETRY AND BRIDGELAND STABILITY CONDITIONS
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
Abstract
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.
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.
Organisations
People |
ORCID iD |
| Christian Boehning (Principal Investigator) |
Publications
Böhning C
(2021)
Prelog Chow groups of self-products of degenerations of cubic threefolds
in European Journal of Mathematics
Böhning C
(2022)
Prelog Chow rings and degenerations
in Rendiconti del Circolo Matematico di Palermo Series 2
Fantini L
(2023)
Triangulations of non-archimedean curves, semi-stable reduction, and ramification
in Annales de l'Institut Fourier
Poineau J
(2022)
Schottky spaces and universal Mumford curves over $${\mathbb {Z}}$$
in Selecta Mathematica
| Description | The aim of this project is to attack long-standing problems in arithmetic geometry: Szpiro's conjecture, originally formulated as an inequality between arithmetic quantities (conductor and discriminant) associated with an elliptic curve over a number field and proved to be equivalent to the widely celebrated "abc conjecture". Szpiro proves a variant of the question over function fields, where the main object of study becomes a family of curves. The first issue one incurs when transposing Szpiro's proof to number fields is that it is not clear what a family should be in this setting, if we require it to be compact. Our key findings were made in the process of adapting to this scenario the main theory of compactification of families over the ring of integers of a number field: Arakelov geometry. In our investigation, we took a two-fold approach: (1) The first one uses Berkovich geometry, a relatively new approach to non-Archimedean analytic spaces that has proven to be a powerful tool for studying various problems in algebraic geometry and number theory, including the arithmetic of elliptic curves. In this framework, a theory of analytic spaces over a number field has been developed by Jérôme Poineau. Our first key finding is that Szpiro's conjecture can be reformulated in terms of this new theory: The Berkovich spectrum M(Z) of the integers (or more generally a ring of integers R in some number field) connects the Archimedean and non-Archimedean places through more general seminorms; thus there arises the possibility of using more genuinely analytic techniques (borrowed from the classical theory of complex analytic spaces) to solve our problem. A simple instance of this phenomenon is the interpretation of the product formula in terms of harmonicity of functions in a neighbourhood of the trivial norm in M(Z), a much more subtle example the recent seminal work on uniform bounds on the number of common images in P^1 of torsion points of elliptic curves; this has been improved on by Jerome Poineau, using continuity results of measures on the fibres of a family of Berkovich spaces over M(R). The two papers co-authored by D. Turchetti are foundational contributions in the theory of Berkovich spaces and help making our approach precise. "Schottky spaces" deals with families of curves of a special type (called Mumford curves) in the context of Berkovich spaces over Z. "Triangulations" proves purely non-Archimedean results about arithmetic invariants of curves, and could be used to clarify the local contributions of non-Archimedean fibers of families of elliptic curves over M(Z). We expect these methods to help answer the longstanding question: what should the Kodaira-Spencer map be in the arithmetic setting? (2) The papers co-authored by C. Boehning, concern prelog Chow groups/log-Chow groups, a hybrid of cycle- and monodromy-theoretic invariants attached to degenerations. These were inspired by work of Bloch-Gillet-Soule, which in turn was motivated by developing a non-archimedean analogue of Arakelov geometry. (Pre-)log Chow groups are also of considerable interest in mirror symmetry, especially the approach by Gross-Siebert and others. |
| Exploitation Route | The ideas developed in the papers on (pre-)log Chow groups are already being further developed by Evgeny Shinder (Sheffield) in the form of Chow-monodromy complexes, unifying ideas from mixed Hodge theory with those from our (pre-)log Chow groups. We expect this to have potentially deep applications to rationality questions and cycle-theoretic questions on varieties. The topic will also be of interest to researchers working in mirror symmetry, and we received feedback to that effect from people working in that area (Bernd Siebert, Mark Gross, Lawrence Barrott, among others). Regarding the Berkovich geometry approach, there has been an increasing interest in the applications of Berkovich spaces to arithmetic geometry, as suggested by the invitation of D. Turchetti to give a mini course on the subject at the Institute for Mathematical Research in Zurich. On that occasion, several researchers manifested their interest in computing arithmetic invariants of curves with these tools, including people in the groups of R. Pink, R. Pandharipande, and A. Kresch. Similar discussions took place with arithmetic geometers in the UK (e.g. V. Dokchitser). |
| Sectors | Education,Other |
| Description | As already described in the section on Key Findings, both the work on Berkovich geometry and on (pre-)log Chow groups is starting to have transformative impact within the respective academic communities (algebraic geometers working on mirror symmetry and rationality questions, on the one hand, and arithmetic geometers using techniques of non-archimedean geometry in their work on the other). In addition to what was already mentioned in Key Findings, the paper "Schottky spaces" by Turchetti et al. has become a starting point for seminal new research directions in arithmetic dynamical systems, cf. the forthcoming "Patterson-Sullivan measures on the Berkovich projective line" by Vlere Mehmeti et al. |
| First Year Of Impact | 2022 |
| Sector | Other |
| Impact Types | Cultural |
| Description | Talk at Warwick Open Day: Maths: Beautiful and Useful, "A remarkable inequality for polynomials in one variable" |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Public/other audiences |
| Results and Impact | This was a talk about the Mason-Stothers proof of the polynomial analogue of Szpiro's conjecture/the abc-conjecture. The intention was to make this highly active and deep area in pure mathematics accessible to a wider public and show its beauty and impact in number theory and beyond. There were a lively discussion and a lot of interaction/questions afterwards, which clearly highlighted the interest and fascination of the audience. |
| Year(s) Of Engagement Activity | 2022 |
| Description | Talk in Warwick Mathematical Society (about a polynomial analogue of the abc conjecture/Szpiro's conjecture) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Undergraduate students |
| Results and Impact | This was a talk and subsequent discussion intended to make some of the open problems and ideas pursued in this research proposal accessible to undergraduate students and members of the general public with an interest in mathematics, especially number theory or geometry. It generated a lot of interest among participants and led to a lively discussion afterwards. |
| Year(s) Of Engagement Activity | 2022 |
| Description | The shape of our world (RI Masterclass) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Schools |
| Results and Impact | This was an online masterclass for high school students on themes related to geometry. I have used part of this to highlight how algebraic geometry plays a role in contemporary research outcomes including those of the current project. |
| Year(s) Of Engagement Activity | 2022 |