  This paper is dedicated to provide theta function representation of
algebro-geometric solutions and related crucial quantities for the modified
Camassa-Holm (MCH) hierarchy through %and studying a algebro-geometric initial
value problem. Our main tools include the polynomial recursive formalism to
derive the MCH hierarchy, the hyperelliptic curve with finite number of genus,
the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type
equations for auxiliary divisors, and the associated trace formulas. With the
help of these tools, the explicit representations of the Baker-Ahhiezer
functions, the meromorphic function, and the algebro-geometric solutions are
obtained for the entire MCH hierarchy.
