In the recent years there have been some spectacular applications of the theory of o-minimality (a branch of Model Theory) to some problems in Diophantine Geometry. It culminated in the unconditional proof of the Andre-Oort conjecture on the Zariski closure of sets of special points on Shimura varieties. We will present ideas and methods surrounding this proof.
In this lecture series I will explain how one can use deformation theory to study derived categories in positive characteristic.
I will start by giving an overview on what does it mean to 'lift' something 'to characteristic 0' and when is this possible. Then I will present a baby example: the study of the Fourier-Mukai partners of products of elliptic curves over algebraically closed fields of characteristic at least 5. After that, I will present Lieblich-Olsson deformation technique which allows us to deform derived equivalence. This is a very versatile tools with many applications (not just in positive characteristic!). I will conclude the series by going over some of these applications in greater details.
In 1985, K. Saito introduced elliptic root systems as root systems belonging to a real vector space F equipped with a symmetric bilinear form I with signature (l, 2, 0). Such root systems are studied in view of simply elliptic singularities which are surface singularities with a regular elliptic curve in its resolution. K. Saito had classified elliptic root systems R with its one-dimensional subspace G of the radical of I, in the case when R/G ⊂ F/G is a reduced affine root system. In our joint work with A. Fialowski and Y. Saito, we have completed its classification; we classified the pair (R,G) whose quotient R/G ⊂ F/G is a non-reduced affine root system. In this talk, we give an overview of elliptic root systems and describe some of the new root systems we have found.
Let E be an elliptic curve defined over ℚ. The ℚ̅-points of E form an abelian group on which the Galois group Gℚ=Gal(ℚ̅/ℚ) acts. The usual Galois representation associated to E captures the action of Gℚ on the points of finite order. However, one could also look at the action of Gℚ on the free part of E(ℚ̅). This infinite-dimensional representation encodes a great deal of interesting arithmetic information. I will state a conjecture concerning this other Galois representation and present supporting evidence from probability theory, Ramsey theory, algebraic geometry, and number theory.
Hilbert's Irreducibility Theorem shows that irreducibility over the field of rationals is 'often' preserved when one specializes a variable in some irreducible polynomial in several variables. I will present a version 'over the ring' for which the specialized polynomial remains irreducible over the ring of integers. The result also relates to the Schinzel Hypothesis about primes in value sets of polynomials: I will discuss a weaker 'relative' version for the integers and the full version for polynomials. The results extend to other base rings than the ring of integers; the general context is that of rings with a product formula.
We review the famous Yomdin-Gromov Lemma about smooth reparametrizations of semialgebraic sets, and state a version of this lemma for holomorphic reparametrizations and semialgebraic sets defined over ℚ. We then introduce o-minimal theory and illustrate how it can deeply impact diophantine geometry.
Let E be an elliptic curve with CM by the imaginary quadratic order OD of negative discriminant D. Given p a prime, if p is inert or ramified in the quadratic field generated by √D then E has supersingular reduction at a(ny) fixed place above p. By a variant of Duke’s equidistribution theorem, as D grows along such discriminants, the proportion of CM elliptic curves with CM by OD whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes p1, . . . ,ps and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank 2 actions on products of locally homogeneous spaces implies that as D grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual pi (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement — obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.
We shall consider elliptic pencils, of which the best-known example is probably the Legendre family Lt: y2=x(x-1)(x-t) where t is a parameter. Given a section P(t) (i.e. a family of points on Lt depending on t) it is an issue to study the set of complex b such that P(b) is torsion on Lb. We shall recall a number of results on the nature of this set. Then we shall present some applications (obtained jointly with P. Corvaja) to elliptical billiards. For instance, if two players hit the same ball with directions forming a given angle in (0,𝞹), there are only finitely many cases for which both billiard trajectories are periodic.
Riffaut (2019) conjectured that a singular modulus of degree h<2 cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results. A joint work with Florian Luca and Amalia Pizarro.
I’ll survey some of the key challenges around the solubility of polynomial Diophantine equations over the integers. While studying individual equations is often extraordinarily difficult, the situation is more accessible if we merely ask what happens on average and if we restrict to the so-called Fano range, where the number of variables exceeds the degree of the polynomial. Indeed, about 20 years ago, it was conjectured by Poonen and Voloch that random Fano hypersurfaces satisfy the Hasse principle, which is the simplest necessary condition for solubility. After discussing related results I’ll report on joint work with Pierre Le Boudec and Will Sawin where we establish this conjecture for all Fano hypersurfaces, except cubic surfaces.
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