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Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

Trace formulas. --- Geometry, Algebraic. --- Harmonic analysis. --- p-adic groups. --- Analyse harmonique (mathématiques) --- Groupes p-adiques. --- Lie groups. --- Lie, Groupes de. --- Formules de trace. --- Trace formulas --- Geometry, Algebraic --- Algebraic geometry --- Geometry --- Formulas, Trace --- Automorphic forms --- Discontinuous groups --- Representations of groups