A conjecture in arithmetic theory of differential equations by Katz N.M.

By Katz N.M.

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2~ s2 n (S2 n ,w) over t h e ground is invariant by symplectic automorphisms of · · · Another natural 1nvar1ant 1s even number A be a k-dimensional ran k wA, where We shall prove that k S. 1c h 1s and 2~ classify the sub- since they are the only independent symplectic invariants of subspaces. 1. 1) is caZZed the skewo~hogonaZ Clearly, dim 'A subspace of A. 2) 1A · For 1nstance, 1· f d~f An 'A k =1, = w( x,x ) {x E A/i(x)wA =0 · 1·1es 1mp = 0} 1A -f 0 -- A c 'A. -oaZ of the subspace A. 2. o) s (A+B) = 5 A n a) 1 B; A c B implies s (A d) n B) 's c 'A; b) = sA + s B.

25) for in F' E f'I E F' • in Then F. f+ (f I' F+. Indeed, the form F, and we may distinguish h-orthonormal bases -/:1 f I ) is a symplectic basis in we shall have a symplectic basis The transformation rr(f 1 ) Sc, it sends on cr also acts on -1:1 w(v,w) h(v,w) is a hermitian metric on fl Sp(S) extends F to = f;, cr(£ 1 ) 5c • (f'I ' -1:1 f') I = r; Similarly, with is then symplectic F', and it commutes with complex conjugation. Therefore, cr E Sp{S), which proves the transitivity property mentioned, Furthermore, it is easy to understand that the isotropy subgroup of F is the unitary group Hence, we get U(F,h) with h given by (2,3,25).

The use of a M. It is chosen adapted basis) also allows for a straightforward checking of the relation 38 Q(L 1 ,L") + Q(L",L" 1 ) for L1 ,L",L" 1 E r'cL). is an affine space. Hence the triple (TCL),::L, Q T(L) then L1 , L" E T(L) Lagrangian subspaaes iff and are transversal. L" of Lagrangian subspaaes of (L,L 1 ) (S,w), there exist aorrrnon transversal. 16. 4 for B ~ T. to both L and L" L1 • =T L n L1 L". d. Let = L n L". 18, [21]. suah that There- is easily seen to be a Then, we have the corresponding space 0 it is clear that L1 which is transversal Lagrangian subspace of S whose intersection with L''.

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