Algebraic_tori_over_fields Algebraic_torus
1 algebraic tori on fields
1.1 multiplicative group of field
1.2 definition
1.3 isogenies
1.4 examples
algebraic tori on fields
in places suppose base field perfect (for example finite or characteristic zero). in general 1 has use separable closures instead of algebraic closures.
multiplicative group of field
if
f
{\displaystyle f}
field multiplicative group on
f
{\displaystyle f}
algebraic group
g
m
{\displaystyle \mathbf {g} _{\mathbf {m} }}
such field extension
e
/
f
{\displaystyle e/f}
e
{\displaystyle e}
-points isomorphic group
e
×
{\displaystyle e^{\times }}
. define algebraic group 1 can take affine variety defined equation
x
y
=
1
{\displaystyle xy=1}
in affine plane on
f
{\displaystyle f}
coordinates
x
,
y
{\displaystyle x,y}
. multiplication given restricting regular rational map
f
2
×
f
2
→
f
2
{\displaystyle f^{2}\times f^{2}\to f^{2}}
defined
(
(
x
,
y
)
,
(
x
′
,
y
′
)
)
↦
(
x
x
′
,
y
y
′
)
{\displaystyle ((x,y),(x ,y ))\mapsto (xx ,yy )}
, inverse restriction of regular rational map
(
x
,
y
)
↦
(
y
,
x
)
{\displaystyle (x,y)\mapsto (y,x)}
.
definition
let
f
{\displaystyle f}
field algebraic closure
f
¯
{\displaystyle {\overline {f}}}
.
f
{\displaystyle f}
-torus algebraic group defined on
f
{\displaystyle f}
isomorphic on
f
¯
{\displaystyle {\overline {f}}}
finite product of copies of multiplicative group.
in other words, if
t
{\displaystyle \mathbf {t} }
f
{\displaystyle f}
-group torus if , if
t
(
f
¯
)
≅
(
f
¯
×
)
r
{\displaystyle \mathbf {t} ({\overline {f}})\cong ({\overline {f}}^{\times })^{r}}
r
≥
1
{\displaystyle r\geq 1}
. basic terminology associated tori follows.
the integer
r
{\displaystyle r}
called rank or absolute rank of torus
t
{\displaystyle \mathrm {t} }
.
the torus said split on field extension
e
/
f
{\displaystyle e/f}
if
t
(
e
)
≅
(
e
×
)
r
{\displaystyle \mathbf {t} (e)\cong (e^{\times })^{r}}
. there unique minimal finite extension of
f
{\displaystyle f}
on
t
{\displaystyle \mathbf {t} }
split, called splitting field of
t
{\displaystyle \mathbf {t} }
.
the
f
{\displaystyle f}
-rank of
t
{\displaystyle \mathbf {t} }
maximal rank of split sub-torus of
t
{\displaystyle \mathbf {t} }
. torus split if , if
f
{\displaystyle f}
-rank equals absolute rank.
a torus said anisotropic if
f
{\displaystyle f}
-rank zero.
isogenies
an isogeny between algebraic groups surjective morphism finite kernel; 2 tori said isogenous if there exists isogeny first second. isogenies between tori particularly well-behaved: isogeny
ϕ
:
t
→
t
′
{\displaystyle \phi :\mathbf {t} \to \mathbf {t} }
there exists dual isogeny
ψ
:
t
′
→
t
{\displaystyle \psi :\mathbf {t} \to \mathbf {t} }
such
ψ
∘
ϕ
{\displaystyle \psi \circ \phi }
power map. in particular being isogenous equivalence relation between tori.
examples
over algebraically closed field there isomorphism unique torus of given rank.
over field of real numbers
r
{\displaystyle \mathbb {r} }
there (up isomorphism) 2 tori of rank 1:
the split torus
r
×
{\displaystyle \mathbb {r} ^{\times }}
;
the compact form, can realised unitary group
u
(
1
)
{\displaystyle \mathbf {u} (1)}
or special orthogonal group
s
o
(
2
)
{\displaystyle \mathrm {so} (2)}
. anisotropic torus. lie group, isomorphic 1-torus
t
1
{\displaystyle \mathbf {t} ^{1}}
, explains picture of diagonalisable algebraic groups tori.
any real torus isogenous finite sum of two; example real torus
c
×
{\displaystyle \mathbb {c} ^{\times }}
doubly covered (but not isomorphic to)
r
×
×
t
1
{\displaystyle \mathbb {r} ^{\times }\times \mathbb {t} ^{1}}
. gives example of isogenous, non-isomorphic tori.
over finite field
f
q
{\displaystyle \mathbb {f} _{q}}
there 2 rank-1 tori: split one, of cardinality
q
−
1
{\displaystyle q-1}
, , anisotropic 1 of cardinality
q
+
1
{\displaystyle q+1}
. latter can realised matrix group
{
(
t
d
u
u
t
)
:
t
,
u
∈
f
q
,
t
2
−
d
u
2
=
1
}
⊂
s
l
2
(
f
q
)
{\displaystyle \left\{{\begin{pmatrix}t&du\\u&t\end{pmatrix}}:t,u\in \mathbb {f} _{q},t^{2}-du^{2}=1\right\}\subset \mathrm {sl} _{2}(\mathbb {f} _{q})}
.
more generally, if
e
/
f
{\displaystyle e/f}
finite field extension of degree
d
{\displaystyle d}
weil restriction
e
{\displaystyle e}
f
{\displaystyle f}
of multiplicative group of
e
{\displaystyle e}
f
{\displaystyle f}
-torus of rank
d
{\displaystyle d}
,
f
{\displaystyle f}
-rank 1 (note restriction of scalars on inseparable field extension yield commutative algebraic group not torus). kernel
n
e
/
f
{\displaystyle n_{e/f}}
of field norm torus, anisotropic , of rank
d
−
1
{\displaystyle d-1}
.
f
{\displaystyle f}
-torus of rank 1 either split or isomorphic kernel of norm of quadratic extension. 2 examples above special cases of this: compact real torus kernel of field norm of
c
/
r
{\displaystyle \mathbb {c} /\mathbb {r} }
, anisotropic torus on
f
q
{\displaystyle \mathbb {f} _{q}}
kernel of field norm of
f
q
2
/
f
q
{\displaystyle \mathbb {f} _{q^{2}}/\mathbb {f} _{q}}
.
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