Seminar of Algebra

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Elliptic curves and locally cyclic Galois extensions

Speaker:
Sara Arias de Reyna Domínguez (Universidad de Sevilla)
Email:
sara_arias@us.es
Location:
Departamento de Álgebra
Date:
Tue, 1 oct 2024 12:30
Soy profesora titular en el departamento de Álgebra de la Universidad de Sevilla. Mi campo de investigación es la teoría de números y la geometría aritmética. Me interesan especialmente las representaciones de Galois.

This talk addresses the problem of finding Galois extensions of the field of rational numbers with prefixed Galois group $G$, such that the decomposition group at each prime is cyclic. Such extensions are called locally cyclic. In the case when the group $G$ is solvable, it is known that there exist locally cyclic Galois extensions of $\mathbf{Q}$ with Galois group $G$. When $G$ is not solvable, the existence of these extensions is widely open. We give a construction of locally cyclic extensions with Galois group GL$(2,p)$ for each prime number $p$, using the Galois representations attached to the $p$-torsion points of elliptic curves. This is joint work with Joachim K\"{o}nig.