Hermitian Analogue of a Theorem of Springer - Chennai ...

HERMITIAN ANALOGUE OF A THEOREM OF SPRINGER 787
PROPOSITION 4.1. Let k be a p-adic field containing all pth roots of unity
and let l be a ramified quadratic extension of k. Then there exists a central
di�ision algebra o�er lŽŽ t.. of degree a power of p, which has an in�olution of
the second kind.
Proof. Let A be a central simple algebra over lŽŽ t.. which is in
Ž ŽŽ ... 2 Br l t . Let � denote the image of A in H ŽŽŽ.. l t , � .
p p . Then A
supports an involution of the second kind which is identity on kŽŽ t.. if and
2 only if N Ž � . � 0 �S, p. 309, Theorem 9.5 �.
Therefore to prove the
lŽŽ t..� kŽŽ t..
proposition, it is enough to show that N 2 lŽŽ t..� kŽŽ t.. is not injective. We have
the following commutative diagramme
2 Ž . 2 ŽŽŽ.. . 1 Ž .
0 � H l, � � H l t , � � H l, ��p� �0
p p
�
2 2 1
Nl�k NlŽŽ t..� kŽŽ t.. Nl�k
�
Ž . Ž ŽŽ .. . Ž .
2 2 1
0 �H k, �p �H k t , �p �H k, ��p� �0
with exact rows. By the local class field theory N 2 is an isomorphism.
l � k
Therefore, by the snake lemma, we have
�
ker N 2 � ker N 1 Ž lŽŽ t..� kŽŽ t.. . Ž l� k . .
Thus, it is enough to show that N 1 l � k is not injective. Since k contains all
1Ž . 1Ž . p
pth roots of unity, we have H k, ��p� � H k, � � k*�k* . Similarly
p
1Ž . p 1
H l, ��p� � l*�l* and the map Nl� k corresponds to the map N:
l*�l* p � k*�k* p induced by the norm map N l � k:
l* � k*. Since l�k is
ramified, the order of the finite group l*�l* p is greater than that of
k*�k* p . In particular, the map N is not injective. This proves the proposition.
EXAMPLE 4.2. Let p be an odd prime number and let k be a p-adic
field. Let l be a ramified quadratic extension of k. Let D be a division
algebra over lŽŽ t.. of degree a power of p, with an involution � of the
second kind. Such a division algebra exists by 4.1. Let � � k* be such that
� is not a norm from l*. Let h � ² 1, ��: be the rank 2 hermitian form
over Ž D, � . .
THEOREM 4.3. Let k, l, D, � , and h be as in 4.2. Then h is anisotropic
o�er D, and there exist finite extensions K and K of kŽŽ t.. 1 2 such that K1 is a
quadratic extension, K 2 is an extension of odd degree, and h is isotropic o�er
D � K for i � 1, 2. In particular, the group UŽ h. kŽŽ t.. i
of isometries of h is
anisotropic o�er kŽŽ t.. and isotropic o�er K and K .
1 2
Proof. Suppose that h is isotropic over D. Then, there exists u � D*
Ž .
such that � � u� u . By taking the reduced norm on both sides, we get