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This paper is a sequel to our previous work, where we proved the ``modularitytheorem'' for algebraic Witt vectors over imaginary quadratic fields. Thistheorem states that, in the case of imaginary quadratic fields $K$, thealgebraic Witt vectors over $K$ are precisely those generated by the modularvectors whose components are given by special values of deformation family ofFricke modular functions; arithmetically, this theorem implies certaincongruences between special values of modular functions that are notnecessarily galois conjugate. In order to take a closer look at this modularitytheorem, the current paper extends it to the case of CM fields. The mainresults include (i) a construction of algebraic Witt vectors from specialvalues of deformation family of Siegel modular functions on Siegel upper-halfspace given by ratios of theta functions, and (ii) a galois-theoreticcharacterization of which algebraic Witt vectors arise in this modular way,intending to exemplify a general galois-correspondence result which is alsoproved in this paper.

Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions