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10) for any φ G Z 2 ^ 7 1 " 1 ) . Clearly, T+(fc)* : ^ ( S 7 1 " 1 ) —+ £ * ( R n ) . 11) The main result of this section is the following representation theorem. T h e o r e m 4 . 3 . Let k G C + \ { 0 } . The following equivalent. 1) 50 AGMON in R n such that eik^u(x) G B*(Rn). 12) for some ^ Ε ΐ ^ " 1 ) . s. of (4-12). Then the map: (4. *(R n ). 15) for all k G C+\{0}, for all k G C+, and ||β" |β| «,ΙΙι»·(»·> > ^ 2 (]k\) II^IU-C*—») /or * G R\{0}. 4. i/ie trace map: B(TLn) 3 u —» T+(fc)u G L 2 ( 5 n - a ) is a surjective map for every k G C+\{0}.

34 ADLER 92 t+ t~ τ L 2 93 l 94 f- τ 2 Figure 3. T h e o r e m 7. , with period matrix (\ U 0 a b\ 2 b c)> T Im fa b\ {b n cj>0' by adjoining at infinity two curves £)+ and D-, each of genus 3, intersecting transversally in 4 points. The divisor D+ + D- is very ample, with the following linear system _ . _ _ . , 2 2 —7773(777? + m ? ) L(L>+ + D-) = {1,7711,7772,7713,73,771! + 7772, + 777173,7713(771271 - 777272) + 27273} providing the embedding of T2 into P 7 . The vector fields XQ1 and XQ4 are the holomorphic vector fields ofT2 and each of them is doubly tangent to D+ and D- at two points tf1 on each curve as depicted in Figure 4.

3) z = 1 , . . , k + m} ' the latter being a basis. Then increase i = 1 , 2 , . . until for the first time (i = i 0 ) , t h e image Φ ^ , ^ Σ ΰ α ) in P N t satisfies the following* genus(V) = dimension of space of holomorphic top forms of V. 4), with y G Φ(^4) and finite implying y G Φ(·^)? 5) denote L^ and JVt· by L(D) and JV; also let D = $L{D). A S the first two conditions are almost automatic, the last one requires genuine attention. The motivation of this crucial step is that ultimately a divisor on an Abelian variety must satisfy the Adjunction formula.

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