The discriminant quotient formula for global fields

Moshe Jarden; Gopal Prasad

doi:10.1007/BF02698842

Jarden, Moshe; Prasad, Gopal

2007-08-30 00:00:00

APPENDIX The discr;mina-t quotient formula for global fields by Moshe JARDEN and Gopal PRASAD We shall use the notation introduced in w 0, however, in the following t will be an arbitrary finite separable extension of k and if k is a number field, we will now let A denote its ring of integers and B that oft. If k is a global function field, let f be its field of constants and I be that of g; q, (resp. qt) is then the cardinality of f (resp. I). For a place v of k (resp. w of g), k, (resp. g~) will denote the completion of k (resp. t) at v (resp. w). If v is nonarchimedean and k is a number field, then A, (resp. B.) will denote the closure ofA (resp. B) in k, (resp. t~) ; A, is the same as the ring denoted by 0, earlier. ] I~ will denote the usual absolute value on Q, and for each rational prime p, I I, the p-adic absolute value. For v e VI, the absolute value I [, extends to the fractional ideals of k if k is a number field and to the divisors of k if k is a function field. A.1. In case k is a number field, let b(A/Z), b(B/Z) be the discriminants of A/Z, B/Z respectively ([10: w 4]), and D, = I b(A[Z)loo, Dt = I b(B/Z)I=. The relative discriminant b(t]h) of ilk is by definition the discriminant b(B/A) of B/A ([10: w 4]), it is an ideal in A. A.2. The group of divisors of function fields will be written multiplicatively. Let K be a global function field. If a = IIa~ is the prime factorization of a divisor a of K, then its degree, to be denoted degK(a), is defined by ~'~ = n l 0, l; 1, (1) where qx is the cardinality of the field of constants of K. The discriminant D~ of K is by definition equal to ~-2, where g~ is the genus of K. If L is a finite separable extension of K, then ~)(L/K) will denote the different of L/K (see [8: Chapter IV, w 8] for the definition of the different). The relative discri- minant b(L/K) is by definition the divisor N~(~(L/K)) of K. A.3. For a place w oft lying over a nonarchimedean place v of k, let b(gJk,) be the relative discriminant of gw/k,. Then r is unramified if and only if b(t~/k,) is trivial. The v-component of the discriminant b(/[k) is 1-I~l , b(tJk,) and [ b(//k)[, = II,,io [ b(t,~]k,)[,; see [10: w 4, Proposition 5], [14: p. 463]. 116 GOPAL PRASAD Theorem A. -- Let t be a finite separable extension of k. Then Dt/D~ ,:~' = H H I b(t=lk,)l; -I. (2) ,~Vf wl, Proof. ~ Number fields and function fields will be treated separately. (i) k is a number field. We use the following relation for the relative discriminants of the ring of integers ([10: w 4, Proposition 7 (ii)]) b (B/Z)/b = N /Q(b (B/A)). (3) Taking thc absolutc value of both sidcs of the abovc, wc obtain DtfDk t:+' = ] Nk,Q(b(t[k))Is = II[N~/Q(b(t[k))]~l by the product formula (0.1) = 1-[ II [ b(t/k)];x (by [7: Theorem in w I1]) = II IlJb(tdk,)[: I (cf. A.3). (ii) k is a function field*. Let k' = Ik. Then k' and t have the same field of constants, the genus of k' equals that of k ([9: p. 132, Theorem 2]) and the different ~)(k'/k) is trivial. Theorem 8 of [8: Chapter IV] implies then that ~(t[k') = ~(t[k). The Riemann-Hurwitz formula for Ilk' ([8: p. 106, Corollary 2]) gives 2gt -- 2 = [t : k'] (2gk. -- 2) + degt(~(t[k')). (4) By a result on p. 110 of [9], we have [I: t] degt(~(t/k) ) = degk(b(t[k) ), since b(t[k) = Nt/k(~(t[k)). Now multiplying (4) by [I:~] we obtain [l: ][] (2gt -- 2) = [t:k] (2gk -- 2) + deg,(b(t/k)). As qt = A t:n, this leads to & = (5) By (1) and the last result of A.3, ~,~t~k,, __ l-I, II,i , ]b(tjk,)]~-t, formula (2) follows therefore from (5). REFERENCES [1] A. BORI~L, Some finiteness properties of adele groups over number fields, Publ. Math. LH.E.S., 16 (1963), 5-30. [2] A. BORm., Linear Algebraic groups, New York, W. A. Benjamin (1969). [3l A. BomzL and J. de SmBE~rrx~a., Les sous-groupcs ferm~ de rang maximum des groupes de Lie clos, Comment. Math. Hale., 23 (1949), 200-221. * We are indebeted to W.-D. Geyer for a simplification of an earlier version of the proof in this case. VOLUMES OF S-ARITHMETIC QUOTIENTS OF SEMI-SIMPLE GROUPS 117 [4] A. BOREL and G. PRASAD, Finiteness theorems for discrete subgroups of bounded covolume in semi-slmplc groups, P~I. Math. LH.E.S., 69 (1989), 119-171. [5] N. Bou~ara, Groupr et Alg~bres de Lie, chapitrcs IV, Vct VI, Paris, Hcrmarm (1968). [6] F. BRU~rAT and J. TITS, Groupes rdductifs sur un corps local, I, PubL Math. I.H.E.&, 41 (1972), 5-251 ; II, ibid., 60 (1984), 5-184. [7] J. W. S. CASSELS, Global fields, in Algebraic number theory (cd. J. W. S. CAsssLs and A. FR6HLmH), London, Academic Press (1967), 42-84. [8] C. CHEVALLZY, In~oduction to the theory of algcbralc functions of one variable, A.M.S. Math. Suroqys, Number VI (1951). [9] M. DEURING, Lectures on the theory of algebraic functions of one variable, springer-Verlag Lecture Notes Math., 814 (1973). [10] A. Fa6macx~, Local Fields, in Algebraic number theory (ed. J. W. S. CASSELS and A. Fa6macH), London, Aca- demic Press (1967), 1-41. [11] G. HAm)Za, Minkowskische Redukdonstheorie fiber Funktionenk6rpem, Itwentiones Math., 7 (1969), 33-54. [12] G. HARD~R, A Gauss-Bormet formula for discrete arithmetically defined groups, Arm. Sci. mole Norm. Sup., Paris, 4 (1971), 409-455. [13] G. HARDgR, Chevalley groups over function fields and automorphic forms, Ann. Math., 100 (1974), 249-306. [14] H. HAssE, Number theo~, Berlin, Springer-Verlag (1980). [15] H. JACQtmT and R. P. Automorphlc forms on GL(2), Springer-Verlag Lecture Notes Math., 114 (1970). [16] M. KI',~ER, Hasse principle for H t of simply comxected groups, Proc. A.M.S. Syrup. Pure Math., 9 (1966), 159-163. [17] R. Ko~rrz, Tamagawa numbers, Arm. Math., 127 (1988), 629-646. [18] K. F. LAX, Tamagawa number of reductive algebraic groups, Compos. Math., 41 (1980), 153-188. [19] R.P.I..ANOt.A~mS, The volume ofthe fundamental domain for some arithmetical subgroups of Cbevalley groups, Prof. A.M.S. Syrup. Pure Math., 9 (1966), 143-148. [20] I. G. MACDONALD, The volume of a compact Lie group, Inventioues Math., 58 (1980), 93-95. [21] G. A. MAaotrtas, Cobounded subgroups in algebraic groups over local fields, Fumtional Anal. Appl., 11 (1977), 45-57. [22] J. G. M. MAas, Los nombrcs de T~magawa de certains groupes exceptiormels, Bull. Sot. Math. Frame, (1966), 97-140. [23] J. G. M. MARS, The Tamagawa number of SAn, Ann. Math., 89 (1969), 557-574. [24] J'. Oma~RL~-, Nombrcs de Tamagawa, Inventiones Math., 78 (1984), 13-88. [25] T. O~o, On algebraic groups and discontinuous groups, Nagoya Math. J., 27 (1966), 279-322. [26] T. Ouo, On Tamagawa numbers, Prof. A.M.S. Syrup. Pure Math., 9 (1966), 122-132. [27] G. PRAS~, Strong approximation, Ann. Math., 105 (1977), 553-572. [28] G. PRAS~ and M. S. Rxom~A~, Topological central extensions of semi-simple groups over local fields, Ann. Math., 119 (1984), 143-268. [29] J.-P. S~ZRRE, Lie algebras andLie groups, New York, W. A. Benjamin (1965). [30] T. A. SPamO~R, Reductive groups, Prof. A.M.S. Syrup. Pure Math., 83 (1979), Part I, 3-27. [31] R. STZX~mERO, Regular elements of semi-simple algebraic groups, Publ. Math. LH.E.S., 25 (1965), 49-80, [32] J. Trrs, Classification of algebraic semi-simple groups, Prof. A.M.S. Syrup. Pure Math., 9 (1966), 33-62. [33] J. Trrs, Reductive groups over local fields, Proc. A.M.S. Syrup. Pure Math., 88 (1979), Part I, 29-69. [34] A. WxIL, Addles and algebraic groups, Boston, Birkl~user (1982). Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400 005 India Manuscrit re~u le 24 .~in 1988. LANGLAND$,

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The discriminant quotient formula for global fields

$APPENDIX The discr;mina-t quotient formula for global fields by Moshe JARDEN and Gopal PRASAD We shall use the notation introduced in w 0, however, in the following t will be an arbitrary finite separable extension of k and if k is a number field, we will now let A denote its ring of integers and B that oft. If k is a global function field, let f be its field of constants and I be that of g; q, (resp. qt) is then the cardinality of f (resp. I). For a place v of k (resp. w of g), k, (resp. g~) will denote the completion of k (resp. t) at v (resp. w). If v is nonarchimedean and k is a number field, then A, (resp. B.) will denote the closure ofA (resp. B) in k, (resp. t~) ; A, is the same as the ring denoted by 0, earlier. ] I~ will denote the usual absolute value on Q, and for each rational prime p, I I, the p-adic absolute value. For v e VI, the absolute value I [, extends to the fractional ideals of k if k is a number field and to the divisors of k if k is a function field. A.1. In case k is a number field, let b(A/Z), b(B/Z) be the discriminants of A/Z, B/Z respectively ([10: w 4]), and D, = I b(A[Z)loo, Dt = I b(B/Z)I=. The relative discriminant b(t]h) of ilk is by definition the discriminant b(B/A) of B/A ([10: w 4]), it is an ideal in A. A.2. The group of divisors of function fields will be written multiplicatively. Let K be a global function field. If a = IIa~ is the prime factorization of a divisor a of K, then its degree, to be denoted degK(a), is defined by ~'~ = n l 0, l; 1, (1) where qx is the cardinality of the field of constants of K. The discriminant D~ of K is by definition equal to ~-2, where g~ is the genus of K. If L is a finite separable extension of K, then ~)(L/K) will denote the different of L/K (see [8: Chapter IV, w 8] for the definition of the different). The relative discri- minant b(L/K) is by definition the divisor N~(~(L/K)) of K. A.3. For a place w oft lying over a nonarchimedean place v of k, let b(gJk,) be the relative discriminant of gw/k,. Then r is unramified if and only if b(t~/k,) is trivial. The v-component of the discriminant b(/[k) is 1-I~l , b(tJk,) and [ b(//k)[, = II,,io [ b(t,~]k,)[,; see [10: w 4, Proposition 5], [14: p. 463]. 116 GOPAL PRASAD Theorem A. -- Let t be a finite separable extension of k. Then Dt/D~ ,:~' = H H I b(t=lk,)l; -I. (2) ,~Vf wl, Proof. ~ Number fields and function fields will be treated separately. (i) k is a number field. We use the following relation for the relative discriminants of the ring of integers ([10: w 4, Proposition 7 (ii)]) b (B/Z)/b = N /Q(b (B/A)). (3) Taking thc absolutc value of both sidcs of the abovc, wc obtain DtfDk t:+' = ] Nk,Q(b(t[k))Is = II[N~/Q(b(t[k))]~l by the product formula (0.1) = 1-[ II [ b(t/k)];x (by [7: Theorem in w I1]) = II IlJb(tdk,)[: I (cf. A.3). (ii) k is a function field*. Let k' = Ik. Then k' and t have the same field of constants, the genus of k' equals that of k ([9: p. 132, Theorem 2]) and the different ~)(k'/k) is trivial. Theorem 8 of [8: Chapter IV] implies then that ~(t[k') = ~(t[k). The Riemann-Hurwitz formula for Ilk' ([8: p. 106, Corollary 2]) gives 2gt -- 2 = [t : k'] (2gk. -- 2) + degt(~(t[k')). (4) By a result on p. 110 of [9], we have [I: t] degt(~(t/k) ) = degk(b(t[k) ), since b(t[k) = Nt/k(~(t[k)). Now multiplying (4) by [I:~] we obtain [l: ][] (2gt -- 2) = [t:k] (2gk -- 2) + deg,(b(t/k)). As qt = A t:n, this leads to & = (5) By (1) and the last result of A.3, ~,~t~k,, __ l-I, II,i , ]b(tjk,)]~-t, formula (2) follows therefore from (5). REFERENCES [1] A. BORI~L, Some finiteness properties of adele groups over number fields, Publ. Math. LH.E.S., 16 (1963), 5-30. [2] A. BORm., Linear Algebraic groups, New York, W. A. Benjamin (1969). [3l A. BomzL and J. de SmBE~rrx~a., Les sous-groupcs ferm~ de rang maximum des groupes de Lie clos, Comment. Math. Hale., 23 (1949), 200-221. * We are indebeted to W.-D. Geyer for a simplification of an earlier version of the proof in this case. VOLUMES OF S-ARITHMETIC QUOTIENTS OF SEMI-SIMPLE GROUPS 117 [4] A. BOREL and G. PRASAD, Finiteness theorems for discrete subgroups of bounded covolume in semi-slmplc groups, P~I. Math. LH.E.S., 69 (1989), 119-171. [5] N. Bou~ara, Groupr et Alg~bres de Lie, chapitrcs IV, Vct VI, Paris, Hcrmarm (1968). [6] F. BRU~rAT and J. TITS, Groupes rdductifs sur un corps local, I, PubL Math. I.H.E.&, 41 (1972), 5-251 ; II, ibid., 60 (1984), 5-184. [7] J. W. S. CASSELS, Global fields, in Algebraic number theory (cd. J. W. S. CAsssLs and A. FR6HLmH), London, Academic Press (1967), 42-84. [8] C. CHEVALLZY, In~oduction to the theory of algcbralc functions of one variable, A.M.S. Math. Suroqys, Number VI (1951). [9] M. DEURING, Lectures on the theory of algebraic functions of one variable, springer-Verlag Lecture Notes Math., 814 (1973). [10] A. Fa6macx~, Local Fields, in Algebraic number theory (ed. J. W. S. CASSELS and A. Fa6macH), London, Aca- demic Press (1967), 1-41. [11] G. HAm)Za, Minkowskische Redukdonstheorie fiber Funktionenk6rpem, Itwentiones Math., 7 (1969), 33-54. [12] G. HARD~R, A Gauss-Bormet formula for discrete arithmetically defined groups, Arm. Sci. mole Norm. Sup., Paris, 4 (1971), 409-455. [13] G. HARDgR, Chevalley groups over function fields and automorphic forms, Ann. Math., 100 (1974), 249-306. [14] H. HAssE, Number theo~, Berlin, Springer-Verlag (1980). [15] H. JACQtmT and R. P. Automorphlc forms on GL(2), Springer-Verlag Lecture Notes Math., 114 (1970). [16] M. KI',~ER, Hasse principle for H t of simply comxected groups, Proc. A.M.S. Syrup. Pure Math., 9 (1966), 159-163. [17] R. Ko~rrz, Tamagawa numbers, Arm. Math., 127 (1988), 629-646. [18] K. F. LAX, Tamagawa number of reductive algebraic groups, Compos. Math., 41 (1980), 153-188. [19] R.P.I..ANOt.A~mS, The volume ofthe fundamental domain for some arithmetical subgroups of Cbevalley groups, Prof. A.M.S. Syrup. Pure Math., 9 (1966), 143-148. [20] I. G. MACDONALD, The volume of a compact Lie group, Inventioues Math., 58 (1980), 93-95. [21] G. A. MAaotrtas, Cobounded subgroups in algebraic groups over local fields, Fumtional Anal. Appl., 11 (1977), 45-57. [22] J. G. M. MAas, Los nombrcs de T~magawa de certains groupes exceptiormels, Bull. Sot. Math. Frame, (1966), 97-140. [23] J. G. M. MARS, The Tamagawa number of SAn, Ann. Math., 89 (1969), 557-574. [24] J'. Oma~RL~-, Nombrcs de Tamagawa, Inventiones Math., 78 (1984), 13-88. [25] T. O~o, On algebraic groups and discontinuous groups, Nagoya Math. J., 27 (1966), 279-322. [26] T. Ouo, On Tamagawa numbers, Prof. A.M.S. Syrup. Pure Math., 9 (1966), 122-132. [27] G. PRAS~, Strong approximation, Ann. Math., 105 (1977), 553-572. [28] G. PRAS~ and M. S. Rxom~A~, Topological central extensions of semi-simple groups over local fields, Ann. Math., 119 (1984), 143-268. [29] J.-P. S~ZRRE, Lie algebras andLie groups, New York, W. A. Benjamin (1965). [30] T. A. SPamO~R, Reductive groups, Prof. A.M.S. Syrup. Pure Math., 83 (1979), Part I, 3-27. [31] R. STZX~mERO, Regular elements of semi-simple algebraic groups, Publ. Math. LH.E.S., 25 (1965), 49-80, [32] J. Trrs, Classification of algebraic semi-simple groups, Prof. A.M.S. Syrup. Pure Math., 9 (1966), 33-62. [33] J. Trrs, Reductive groups over local fields, Proc. A.M.S. Syrup. Pure Math., 88 (1979), Part I, 29-69. [34] A. WxIL, Addles and algebraic groups, Boston, Birkl~user (1982). Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400 005 India Manuscrit re~u le 24 .~in 1988. LANGLAND$,$

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The discriminant quotient formula for global fields

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