The notion of pseudo-representations was initially introduced for group algebras by Wiles (for GL2) and by Taylor (for GLd) in order to construct Galois representations associated to certain automorphic forms. Chenevier proposed an alternative theory of "determinant laws" which extends Wiles and Taylor’s definition to arbitrary rings. This theory has proved to be useful in the study of congruences between automorphic forms and in the deformation theory of residually reducible Galois representations. In this talk, I will present my joint work with Julian Quast on symplectic determinant laws, which adapts Chenevier’s framework to the symplectic group GSp2d. I will give the definition and highlight some of its key properties, and then explain its connection to Geometric Invariant Theory. In particular, I will show that in characteristic zero, the space of symplectic determinant laws recovers the GIT quotient of the space of symplectic representations by the conjugation action.
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