PhD Defence of H.J. Smit MSc

Relating Galois representations via L-series and dynamical systems

In day-to-day life we often work with incomplete information: a doctor trying to diagnose a patient by doing a blood count, a meteorologist trying to predict whether or not it will rain in Utrecht tomorrow at 3:27PM, or a friend trying to figure out your emotions by looking at your face. From scattered pieces of information we try to reconstruct or characterize the system we study and make predictions about it. In order to distinguish two situations it is often useful to look at specific information that is different in both situations.

This thesis tries to accomplish an abstract mathematical variant of this in the setting of number theory. Here we have some objects of interest (Galois representations) and we would like to know whether or not these representations have the same shape. Our way of approaching this is to use associated invariants known as L-series. These are functions that can be calculated from this Galois representation. The results of the thesis (technical details aside) then state that if two Galois representations have enough L-series in common, then the Galois representations must be essentially of the same shape.

Secondly, a similar statement holds in a slightly different situation: number fields (another object of interest in number theory) can be characterized by an associated dynamical system. The strength of this result lies in the fact that this dynamical system only uses "abelian'' information (read: simple information) of the number field. Most previous results required some form of non-abelian input.

Start date and time
End date and time
PhD candidate
H.J. Smit MSc
Relating Galois representations via L-series and dynamical systems
PhD supervisor(s)
prof. dr. G.L.M. Cornelissen