Finiteness theorems for potentially equivalent Galois representations: Extension of Faltings’ finiteness criteria

We study the relationship between potential equivalence and character theory; we observe that of a representation ρ \rho is determined by an equality alttext="m"> m encoding="application/x-tex">m -power alttext="g right-arrow from bar upper T r left-parenthesis rho g Superscript m Baseline right-parenthesis right-parenthesis"> g ↦ T r stretchy="false">( stretchy="false">) encoding="application/x-tex">g\mapsto Tr(\rho (g^m)) for some natural number . Using this, extend Faltings’ finiteness criteria to determine two alttext="script l"> ℓ<!-- ℓ encoding="application/x-tex">\ell -adic, semisimple representations absolute Galois group field, context equivalence. also discuss results twist unramified representations.