Page 432 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 432
NUMBER THEORY (MS-63)
common factors. Silverman conjectured that a similar behaviour should appear in (a large class
of) all algebraic groups. We show, extending work of Ghioca-Hsia-Tucker and Silverman on
elliptic curves, how to prove Silverman’s conjecture over function fields for abelian and split
semi-abelian varieties and some generalizations.
Atkin-Lehner theory for Drinfeld Modular forms
Maria Valentino, maria.valentino@unical.it
University of Calabria, Italy
Let Sk(N ) be the space of cusp forms of level Γ0(N ) with N ∈ Z. Atkin-Lehner theory deals
with the notion of oldforms, namely those coming from a lower level M |N , and newforms,
i.e. the orthogonal complement of the space of oldforms with respect to the Petersson inner
product. Moreover, it is also concerned with the construction of a basis for Sk(N ) made up
by eigenfunctions for the Hecke operators Tn with n prime to N . In this talk we shall present
some recent advances on the analogous theory for Drinfeld modular forms, which are certain
analogues over the function field Fq[T ] of classical modular forms.
The field of iterates of a rational function
Solomon Vishkautsan, wishcow@telhai.ac.il
Tel-Hai Academic College, Israel
Coauthor: Francesco Veneziano
We will discuss the field of definition of a rational function and in what ways it can change
under iteration, in particular when the degree over the base field drops. We present examples
of families of rational functions with the property above, and prove that in the special case of
polynomials, only one of these families is possible. We also explain how this relates to Ritt’s
decomposition theorem on polynomials. Joint work with Francesco Veneziano (SNS Pisa).
430
common factors. Silverman conjectured that a similar behaviour should appear in (a large class
of) all algebraic groups. We show, extending work of Ghioca-Hsia-Tucker and Silverman on
elliptic curves, how to prove Silverman’s conjecture over function fields for abelian and split
semi-abelian varieties and some generalizations.
Atkin-Lehner theory for Drinfeld Modular forms
Maria Valentino, maria.valentino@unical.it
University of Calabria, Italy
Let Sk(N ) be the space of cusp forms of level Γ0(N ) with N ∈ Z. Atkin-Lehner theory deals
with the notion of oldforms, namely those coming from a lower level M |N , and newforms,
i.e. the orthogonal complement of the space of oldforms with respect to the Petersson inner
product. Moreover, it is also concerned with the construction of a basis for Sk(N ) made up
by eigenfunctions for the Hecke operators Tn with n prime to N . In this talk we shall present
some recent advances on the analogous theory for Drinfeld modular forms, which are certain
analogues over the function field Fq[T ] of classical modular forms.
The field of iterates of a rational function
Solomon Vishkautsan, wishcow@telhai.ac.il
Tel-Hai Academic College, Israel
Coauthor: Francesco Veneziano
We will discuss the field of definition of a rational function and in what ways it can change
under iteration, in particular when the degree over the base field drops. We present examples
of families of rational functions with the property above, and prove that in the special case of
polynomials, only one of these families is possible. We also explain how this relates to Ritt’s
decomposition theorem on polynomials. Joint work with Francesco Veneziano (SNS Pisa).
430
