The worthy purpose of this text is to provide a complete, self-contained development of the trace formula and theta inversion formula for SL(2,Z[i])\SL(2,C). Unlike other treatments of the theory, the approach taken here is to begin with the heat kernel on SL(2,C) associated to the invariant Laplacian, which is derived using spherical inversion. The heat kernel on the quotient space SL(2,Z[i])\SL(2,C) is arrived at through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the Gauss transform.</

Gaussians, Spherical Inversion, and the Heat Kernel.- Spherical Inversion on SL2(C).- The Heat Gaussian and Heat Kernel.- QED, LEG, Transpose, and Casimir.- Enter ?: The General Trace Formula.- Convergence and Divergence of the Selberg Trace.- The Cuspidal and Noncuspidal Traces.- The Heat Kernel on ?\G/K.- The Fundamental Domain.- ?-Periodization of the Heat Kernel.- Heat Kernel Convolution on (?\G/K).- Fourier-Eisenstein Eigenfunction Expansions.- The Tube Domain for ??.- The ?/U-Fourier Expansion of Eisenstein Series.- Adjointness Formula and the ?\G-Eigenfunction Expansion.- The Eisenstein-Cuspidal Affair.- The Eisenstein Y-Asymptotics.- The Cuspidal Trace Y-Asymptotics.- Analytic Evaluations.