By Andre Weil
This quantity comprises the unique lecture notes offered by way of A. Weil during which the idea that of adeles was once first brought, at the side of a number of elements of C.L. Siegel’s paintings on quadratic kinds. those notes were supplemented through a longer bibliography, and by means of Takashi Ono’s short survey of next examine. Serving as an advent to the topic, those notes can also supply stimulation for additional learn.
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It is easily seen that H is also the orbit of e under G. m-1 ' Call g the subgroup of G leaving e fixed; it consists of the matr~ ces 1 X= ( 0 (tu tu) X' ( u E Om-1, X' E. G' ) is the transpose of u). If x=(x i )1
Let G and gcG be algebraic groups defined .. over k. 2 is fulfilled. 2. 2 is automatically verified if k is a number-field (Rosenlicht). For the function fields it is easily verified in each of the particular cases treated in Chapters III and IV. 3. 2. Let dx, da, dP be gauge-forms defined over k on G,g,G/g, matching together algebraically. Let (A V)' (~v), (vv) be three sets of factors for k with AV = ~v 'vv' Then: (1) If two of three sets (A V)' (~v), (vv) are sets of convergence factors (for G, g, GIg respectively), so is the third one.
Furtermore if p does not divide the discriminant of D then Do is a maximal order of Dk ' and one has -p p Dk = Mn(k p)' p Since every maximal order of Mn(k p ) is conjugate to Mn (0-p ), there is II ring isomorphism Do ~ Mn(Q) if pf S, S being suitably chosen. -p p As an invariant measure on D*, we can take w = N(x ) -n dx 1... dx 2 n and if pt. l =/ w = p D* p GL Qp x= (x 1·J·), dX = / (0 ) n -p 11 .. 1 ,J IdetXI- n (dX) = / p P GL (0 ) n -p (dX)p dx lJ .. , q = N(p) . From this formula follows that the Ap = 1 - N(p)-1 are a set of convergence factors for D*.
Adeles and Algebraic Groups by Andre Weil