By Seymour Lipschutz

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P 2 C . @2 rx2 @1 / C O. i 1/ C 1/: In particular, we obtain the weak estimates p 1 t2 p2 p p2 p D @2 p 2 t2 p 2 2pC1 D @2 p 2 2p C r@2 @2 C O. p 2 C 2p p 2 2p p 2 3pC1 D @2 2@2 p2 p p 2 2pC1 t2i D @2 1/; i Äp C O. p 2 C 2p C 1/; p 2 2p C O. p 2 C 2p C 1/: p 2 2p C O. p 2 C 2p C 1/ 3; p 2 2p C r@2 @2 p 2 2p C O. p 2 C 3p/ D @2 t2i D O. x2 @1 / C O. x2 @1 / C @2 2/ D 1 C x2 p 1 This proves (2). 2p 2/: C O. p 2 p/ x2 / i D1 D p2 p @2 2 C O. p 2 p/ x2 . p 2 i D1 D p X1 i D1 . i/ . 13. 1/ which commutes with t1 .

T / D p 2. 2/ . / is abelian. 2/ . 1 C x2 //. Proof. 5(2) shows that they are Winter conjugate. 1 proves that there is a bijection of roots and root space dimensions for any two of these tori. 9(1) proves the first statement. 13. To simplify notation we suppress the notion of ‰. 2/ D F t1 . 2 33 and put D i ˛ C jˇ ¤ 0 for some i; j 2 Fp . 1 C u2 /j . rC1/j j. u2 / D i. uk / for k D 0; 1; 2. 1; 2// as an ordinary algebra. 1; 2//. As the action is a restricted one, this gives d Œp D jt1 it2 . rC1/l for 0 Ä r Ä p 1, and argue as above to show that d Œp D t1 .

P/ q1 WD u1 C p X1 . p/ q2 WD u2 C i D1 p X1 . p 1/ . p 1/ u2 C p X1 . q2 /. q2 / D 0, in this case h is abelian. q2 /, 3 h. Then i ; 2 F . h/ for k D 3; 2. h/. u0 / D 0 2. u0 / : i D1 Next we compute these p-th powers. ul / D 0 for l D 0; 1; 2. ul / D 0 for l D 0; 1; 2 if i > 1. Consequently, hp C t2 C . 1; 2//. Then hŒp C t2 C . 0/ . 0/ is closed under Œp-th powers, this is a 1-dimensional torus. 2/ T D F t1 ˚ F t2 ; P2 for toral elements t1 ; t2 . 2/ and s ¤ 0. 2/ and U has dimension p 2 .