# Transseries in difference and differential equations by Kuik G.R.

[-l]*L which is the identity on the subscheme X of points of order 2. 4) 21dlld21 ···Id g. Let c X for ~ + X Given 0 A9,*( 0,20 ) = (d l ,d 2 ,···,d g), di £ N, be the functor on the category of locally noetherian schemes, such that for any locally noetherian scheme S. x Z/dgZ here ~ K(o) = Pd1x ...

I) For any subgroup is a compact Hausdorff space. 22 r c SP29(Z) of finite index, * (ii) (r\Hg )-(r\Hg) has a natural finite stratification. stratum is a quotient of some Hr , r < g, (iii) Each stratum of r~g* Each by a congruence subgroup. has a natural structure of normal analytic space. 10. Applying Cartan's criterion of extension of analytic structure, we can "glue" these strata together to produce a compact * analytic space structure on r\Hg. 1) Let V be a locally compact, second countable Hausdorff space.

N). 21n. 2n) £ for suitable choice of det(Cn+D)~. nn) In this case. t. 2n). nn) . t. 2n). ( 3) vr £ N. 2 1r. va,b £ r -1 Z9 . t. and satisfies the following transformation law v(~~) r(r 2,2r2). r(r 2 ,2r 2) £ e[~J(o. (An+B)(Cn+D» = det(Cn+D)~ e[~J(o,n) for some suitable choice of det(Cn+D)~. £r- £ r -1 g Z 1 e(-rtR,m) e(;J (O,n) zg/zg v m £ r -2Zg 30 Thus (3) is essentially a special case of (2). I The theta constants {e[~]{o,n) a,b £ r-1Zg/Z g} are those used by Igusa in , and the {e[~]{o,n)la £ n-1Z g} are essentially those used by theta constants ~1umford in [Eq.