Algebraic de Rham theory for the relative completion of {\mathrm{SL}}_2(\mathbb{Z}).

We develop an explicit algebraic de Rham theory for the relative completion of {\mathrm {SL}}_2(\mathbb {Z}). This allows the construction of iterated integrals involving modular forms of the second kind, generalizing iterated integrals of holomorphic modular forms that were previously studied by Brown and Manin. These newly constructed iterated integrals provide all 'multiple modular values' defined by Brown. [ABSTRACT FROM AUTHOR]