Sedenion Algebra and the Geometric Langlands Program: A Computational Implementation

We present a complete computational implementation of the Geometric Langlands Program using sedenion algebra, a 16-dimensional associative Clifford-type algebra introduced by Mironov and Mironov. By constructing a parabolic Higgs field with poles at four points on P1 and optimizing its parameters to match the roots with the Hecke eigenvalues ap of the elliptic curve 11.a1, we demonstrate that the spectral curve of the Higgs field precisely reproduces the Dirichlet coefficients of the L-function. We further verify the construction by applying Hecke operators to a sample sheaf, obtaining eigenvalues that exactly match the LMFDB data.