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.