THE BOSONIC STRING MEASURE AT 2 AND 3 LOOPS AND SYMPLECTIC TRANSFORMATIONS OF THE VOLUME FORM

Symplectic modular invariance of the string integral has been verified at genus 2 and 3 using the period matrix coordinatization of moduli s pace. A calculation of the transformation of the product of holomorphi c coordinates Pi(i less than or equal to j)d tau(ij). shows that an ex tra phase arises together with the factor associated with a specific m odular weight; the phase is cancelled in the transformation of the ent ire volume element including the complex conjugate. An argument is giv en for modular invariance of the reggeon measure at genus 12.