# Seminari di Algebra

**The nilpotent decomposition in integral group rings**

Ore 16.00-16.40

**Leo Margolis** (Universidad Autónoma de Madrid)

*Abstract*

Inspired by the Jordan decomposition in group rings Passman and Liu defined nilpotent decomposition: a finite group G has this property, if for any nilpotent element *n* in the integral group ring of G and every central idempotent *e* of the rational group algebra of G the product *e*n* also lies in the integral group ring. It is easy to see that if the rational group algebra of G has at most one matrix component, then G has also nipotent decomposition. Jespers and Sun conjectured that these two properties are in fact equivalent. We will report on some ongoing work on this conjecture which has already uncovered some rather surprising results.

**The classification of planar monomials over fields of order p**^{3}

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Ore 16.50-17.30

**Irene Villa** (Università di Trento)

*Abstract*

Planar functions are functions defined over finite fields of odd characteristics such that their (nonzero) derivatives are permutations. It was conjectured in 1968 by Dembowski and Ostrom that all planar maps have degree two. This conjecture was proven correct for functions defined over prime fields. Moreover, when restricted to monomials, it was proven correct for fields of order *p*^{2} and, when *p*>3, for fields of order *p*^{4} (a counterexample was constructed for the field of order 3^{4}). In this talk, I will present a proof of the conjecture for monomials over fields of order a prime cubed.

**Elliptic Curves over Hasse pairs**

Ore 17.40-18.20

**Daniele Taufer** (KU Leuven)

*Abstract*

Given a prime power *q*, the size *p* of elliptic curves over F*q* lies inside an interval of integers, usually referred to as the Hasse interval. When this *p* is also a prime power, we have a natural “dual” family of curves defined over F*p* and with a group of points of order *q*. We define such a pair (*p,q*), i.e. a pair of prime powers lying inside the Hasse interval of each other, as a Hasse pair, and we investigate the relation between these families of curves. When supersingular curves are involved, we show that the problem of constructing curves over Hasse pairs may be reduced to well-known open problems in number theory. Instead, when only ordinary curves are involved, we prove a perfect correspondence between their isogeny graphs, which highlights how these seemingly different local objects arise from good reductions of the same global curves. This is joint work with Antoine Joux and Eleni Agathocleous.

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