STNB 2015 (29th year)
From the Generalized Fermat Equation to Hilbert modular forms with prescribed inertia types
Presenters
Abstract
In this talk I will report on ongoing work with Lassina Dembele and John Voight. After the proof of Fermat's Last Theorem the modular method to solve Diophantine equations has been generalized by several mathematicians and used to attack many other equations. As a consequence of these efforts the Generalized Fermat Equation $Ax^r + By^q = Cz^p$, where $A$, $B$, $C$ are pairwise coprime integers, became the new focus of attention. In an attempt to study the particular case case $x^{19} + y^{19} = Cz^p$, among other things, we are led to compute huge spaces of (Hilbert) modular forms. To complete this task, following a suggestions of Fred Diamond, we intend to split the space of modular forms into smaller ones by prescribing the inertia types. These smaller spaces should be amenable to computations.
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