By Carl Ludwig Siegel (Auth.)

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**Example text**

On the other hand, let us consider the definition of a discrete group. A group of matrices 99Î with real (or complex) elements is called discrete, if every infinite sequence of different 3JÎ diverges. I t is obvious that a discontinuous group of symplectic matrices is discrete. Let us now prove the con- 25 SYMPLECTIC GEOMETKY. for 3 = i&> an d consequently On the other hand, (56) άω — 2™ \ g) I»"1 Π {dxkidykl) = 2"1 Π (dxkidYkl) with (Yki) — g)"1. )-S we obtain, by (51), (55) and (56), x = Cn(—^ynin+1)/2 where the positive rational number is defined in (53), (»54) and (57) 2n*[(n2 + f dV9 F n)/2]l dv — U (dxjcidYbi) k

By the result of Section 30, only a finite number of non-associate pairs ©, 2) fulfill that inequality. By Lemma 9, there exists a number λ > 1, such that abs (©3χ + 25) > 1 for all ©, 2) and abs (©3 λ + 2)) = 1 for © = ©0, 2) *= 2)0, where ©0 =7^= 0 and ©0, 2)0 is not associate with one of the pairs © s , 2)*. )-1), all conditions (89), (90) and (91) are satisfied for 3λ instead of 3 ; hence 3λ is a point of F. On the other hand, the expression abs (©3x + 2)) attains its minimum 1 for © = S 0 , 2) = 2)0, and consequently there exists a modular transformation 3 * λ = (8Ι3λ + 95) (©3\ + 2))" 1 , such that ©,2) is associate with ©0, 2>o and 3*x is a point of i 7 .

By (72) and Section 25, the group Δ0(ί@, $ ) consists of all matrices 90? of the form (75), such that (76) is satisfied and GTSTOßr1 is integral. On the other hand, the solutions of (76) with integral 9ί& {k = 0, · · ·, 3) in K constitute also a subgroup A 0 (r, s) of the homogeneous symplectic group Ω0· I t follows from the argument at the end of Section 26, that the two groups Δ0(@, φ ) and A 0 (r, s) are commensurable. The problem 36 CARL LUDWIG SIEGEL. of the commensurability of two groups Δ(@, ip) and Δ (©^ §1) is therefore reduced to the corresponding problem for Δ 0 (^ s) and Δ 0 (ΓΙ, $ Ι ) .