Yuri Manin memorial

Abstracts of talks

Anna Cadoret

On toric points of p-adic local systems arising from geometry

For a smooth variety over a number field and a p-adic local systems arising from geometry on it, classical conjectures on algebraic cycles predict that the toric points should fit with the CM points of the associated variation of Hodge structure; in particular, they should have similar properties in terms of sparsity. I will discuss results in this direction. This is a joint work with Jakob Stix.

Kęstutis Česnavičius

The Manin constant and the modular degree

The Manin constant c of an elliptic curve E over Q is the nonzero integer that scales the differential ω determined by the normalized newform f associated to E into the pullback of a Néron differential under a minimal parametrization \phi : X_0(N) ---> E. Manin conjectured that c = ±1 for optimal parametrizations. We show that in general c | deg(\phi), under a minor assumption at 2 and 3, which improves the status of the Manin conjecture for many E. Our core result that gives this divisibility is the containment ω ∈ H^0(X_0(N), Ω), which we establish by combining automorphic methods with techniques from arithmetic geometry (here the modular curve X_0(N) is considered over Z and Ω is its relative dualizing sheaf over Z). To overcome obstacles at 2 and 3, we analyze nondihedral supercuspidal representations of GL_2(Q_2) and exhibit new cases in which X_0(N) has rational singularities over Z. This is joint work with Abhishek Saha and Michalis Neururer.

Soheyla Feyzbakhsh

Moduli spaces of stable vector bundles on curves on K3 surfaces

Consider the moduli space M of stable rank r vector bundles on a curve C with canonical determinant, and let h be the maximum number of linearly independent global sections of these bundles. If C embeds in a K3 surface X, I will show the sublocus M' of M consisting of bundles with h global sections is a smooth projective hyperkahler manifold. As a result, I will prove Mukai's conjecture saying that the K3 surface X containing C can be obtained uniquely out of the curve C.  

Dan Loughran

The arithmetic of cubic surfaces

I will explain some of my protracted love affair with cubic surfaces which began when I first picked up Manin's book on cubic forms.

Loïc Merel

On the conjecture of Harris and Venkatesh for modular forms of weight one

(with E. Lecouturier)

This is part of the vast world of (a variant of) Venkatesh's conjectures about derived Hecke algebras. The situation of modular forms of weight one considered by Harris and Venkatesh might be the simplest significant case. Consider the one-to-one correspondence between newforms f of weight one and two-dimensional, complex, odd, irreducible representation ρ of the absolute Galois group. Consider such f and ρ with coefficients in a subring A of C. Let q be a prime number unramified for ρ (equivalently, not dividing the level of f). Consider the multiplicative group F_q* of the finite field F_q. Harris and Venkatesh attach, up to sign, two elements in the tensor product of F_q* and A:

– From f, without knowing ρ, a pseudo-eigenvalue of the derived Hecke operator at q, 

– From ρ, without knowing f, the localisation at q of a global cohomology class well defined up to a scalar in A.

The conjecture asserts that the second collection is equal to the first, up to an element of A. Original explanations will be given, and the conjecture will be expanded into new directions, based on the theory of modular symbols, as developed by Manin.

Adam Morgan

On the Hasse principle for Kummer varieties  

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov established the Hasse principle for Kummer varieties associated to 2-coverings of a principally polarised abelian variety A, under certain large image assumptions on the Galois action on A[2]. However, their method stops short of treating the case where the image is the full symplectic group, due to the possible failure of the Shafarevich--Tate group to have square order in this case. I will explain work in progress which overcomes this obstruction by combining second descent ideas in the spirit of Harpaz and Smith with new results on the 2-parity conjecture.

Margherita Pagano

Brauer-Manin obstruction coming from primes of good reduction

I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces. I will then explain how the reduction type (in particular, ordinary or non-ordinary good reduction) plays a role. This is work in progress.

Alec Shute

The Hasse principle for polynomials represented by norm forms

A central question in arithmetic geometry asks under what circumstances the Hasse principle holds for the affine equation given by a polynomial equal to a norm form. In this talk, I present results which establish the Hasse principle for a wide new family of polynomials and number fields, which includes polynomials of arbitrarily large degree. The proof makes use of the beta sieve developed by Rosser and Iwaniec, and also has applications to the study of rational points in fibrations. 

Efthymios Sofos

The second moment method for rational points

In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel's Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen [arXiv:2212.10373] that builds on the second-moment method and establishes asymptotics for averages of a general arithmetic function over the values of typical polynomials. The new tools come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer--Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties. 

Olivier Wittenberg

Supersolvable descent for rational points

The by now classical formalism of descent under a torus introduced by Colliot-Thélène and Sansuc in the 1980's admits an analogue in which the torus is replaced with a supersolvable finite group. I will explain this formalism and discuss applications to rational points of homogeneous spaces of linear groups over number fields and to the inverse Galois problem with prescribed norms. This is joint work with Yonatan Harpaz.

Yuri Zarhin

Central simple representations and superelliptic jacobians

Let f(x) be a polynomial of degree at least 5 with complex coefficients and without repeated roots. Suppose that all the coefficients of f(x) lie in a subfield K such that: 

1) K contains a primitive p-th root of unity;

2) f(x) is irreducible over K;

3) the Galois group Gal(f) acts doubly transitively on the set of roots of f(x);

4) the index of every maximal subgroup of Gal(f) does not divide deg(f)-1. 

Then the endomorphism ring of the Jacobian of the superelliptic curve y^p=f(x) is isomorphic to the p-th cyclotomic ring for all primes p>deg(f). We outline the proof, which is based on ideas from representation theory.