Page 428 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 428
NUMBER THEORY (MS-63)
Higher height paining and extensions of mixed Hodge structures
Jose Ignacio Burgos Gil, burgos@icmat.es
ICMAT-CSIC, Spain
Coauthors: Souvik Joy, Greg Pearlstein
The height pairing between algebraic cycles over global fields is an important arithmetic in-
variant. It can be written as sum of local contributions, one for each place of the ground field.
Following Hain, the Archimedean components of the height pairing can be interpreted in terms
of biextensions of mixed Hodge structures. In this talk we will explore how to extend the
Archimedean contribution of the height pairing to higher cycles in the Bloch complex and in-
terpret it as an invariant associated to a mixed Hodge structure. This is joint work with S.
Goswami and G. Pearlstein.
Multiplicative and linear dependence in finite fields and on elliptic curves
modulo primes
Laura Capuano, laura.capuano@polito.it
Politecnico di Torino, Italy
Coauthors: Fabrizio Barroero, Lazlo Mérai, Alina Ostafe, Min Sha
In 2008 Maurin proved that given n multiplicatively independent rational functions ϕ1(x), . . . ,
ϕn(x) ∈ Q(x), there are at most finitely many α ∈ Q such that ϕ1(α), . . . , ϕn(α) satisfy two
independent multiplicative relations. This statement is an instance of more general conjectures
of unlikely intersections over tori made by Bombieri, Masser and Zannier and independently
by Zilber. We consider a positive characteristic variant of this problem, proving that, for suf-
ficiently large primes, the cardinality of the set of α ∈ Fp such that ϕ1(α), . . . , ϕn(α) satisfy
two independent multiplicative relations with exponents bounded by a certain constant K is
bounded independently of K and p. We prove analogous results for products of elliptic curves
and for split semiabelian varieties En × Gmk .
On the variance of the nodal volume of arithmetic random waves
Giacomo Cherubini, cherubini@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics,
Department of Algebra, Czech Republic
Coauthor: Niko Laaksonen
We discuss arithmetic random waves on the d-dimensional torus Rd/Zd. Their zero set and its
v√onluimneRadre(nre≥late2d). to the study of linear correlations of lattice points on the sphere of radius
In this talk we look for bounds on the variance of the nodal volume. The
problem has been solved (in the stronger form of an asymptotic with power saving) in dimension
d = 2, 3 by using a combination of number theory and graph theory. In this seminar we will
explain what is known in dimension d ≥ 4. As the dimension increases, analysis gives the best
results. The main input is a result that follows from the proof of the l2-decoupling conjecture
by Bourgain and Demeter.
426
Higher height paining and extensions of mixed Hodge structures
Jose Ignacio Burgos Gil, burgos@icmat.es
ICMAT-CSIC, Spain
Coauthors: Souvik Joy, Greg Pearlstein
The height pairing between algebraic cycles over global fields is an important arithmetic in-
variant. It can be written as sum of local contributions, one for each place of the ground field.
Following Hain, the Archimedean components of the height pairing can be interpreted in terms
of biextensions of mixed Hodge structures. In this talk we will explore how to extend the
Archimedean contribution of the height pairing to higher cycles in the Bloch complex and in-
terpret it as an invariant associated to a mixed Hodge structure. This is joint work with S.
Goswami and G. Pearlstein.
Multiplicative and linear dependence in finite fields and on elliptic curves
modulo primes
Laura Capuano, laura.capuano@polito.it
Politecnico di Torino, Italy
Coauthors: Fabrizio Barroero, Lazlo Mérai, Alina Ostafe, Min Sha
In 2008 Maurin proved that given n multiplicatively independent rational functions ϕ1(x), . . . ,
ϕn(x) ∈ Q(x), there are at most finitely many α ∈ Q such that ϕ1(α), . . . , ϕn(α) satisfy two
independent multiplicative relations. This statement is an instance of more general conjectures
of unlikely intersections over tori made by Bombieri, Masser and Zannier and independently
by Zilber. We consider a positive characteristic variant of this problem, proving that, for suf-
ficiently large primes, the cardinality of the set of α ∈ Fp such that ϕ1(α), . . . , ϕn(α) satisfy
two independent multiplicative relations with exponents bounded by a certain constant K is
bounded independently of K and p. We prove analogous results for products of elliptic curves
and for split semiabelian varieties En × Gmk .
On the variance of the nodal volume of arithmetic random waves
Giacomo Cherubini, cherubini@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics,
Department of Algebra, Czech Republic
Coauthor: Niko Laaksonen
We discuss arithmetic random waves on the d-dimensional torus Rd/Zd. Their zero set and its
v√onluimneRadre(nre≥late2d). to the study of linear correlations of lattice points on the sphere of radius
In this talk we look for bounds on the variance of the nodal volume. The
problem has been solved (in the stronger form of an asymptotic with power saving) in dimension
d = 2, 3 by using a combination of number theory and graph theory. In this seminar we will
explain what is known in dimension d ≥ 4. As the dimension increases, analysis gives the best
results. The main input is a result that follows from the proof of the l2-decoupling conjecture
by Bourgain and Demeter.
426
