DOI: 10.1007/s00222-004-0342-y
Invent. math. 157, 455–518 (2004)
N´eron models, Lie algebras, and reduction of curves
of genus one
Qing Liu
1
,
Dino Lorenzini
2
,?
,
Michel Raynaud
3
1
CNRS, Laboratoire A2X, Universit´e de Bordeaux I, 351, cours de la Lib´eration, 33405
Talence, France (e-mail:
Qing.Liu@math.u-bordeaux.fr
)
2
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
(e-mail:
lorenzini@math.uga.edu
)
3
Universit´e de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France
(e-mail:
michel.raynaud@math.u-psud.fr
)
Oblatum 28-VIII-2003 & 24-IX-2003
Published online: 9 June 2004 –
Springer-Verlag 2004
Dedicated to John Tate
Let
K
be a discrete valuation ±eld. Let
O
K
denote the ring of integers
of
K
,andle
t
k
be the residue ±eld of
O
K
, of characteristic
p
≥
0. Let
S
:=
Spec
O
K
.Le
t
X
K
be a smooth geometrically connected projective
curve of genus 1 over
K
. Denote by
E
K
the Jacobian of
X
K
t
X
/
S
and
E
/
S
be the minimal regular models of
X
K
and
E
K
, respectively. In this
article, we investigate the possible relationships between the special ±bers
X
k
and
E
k
. In doing so, we are led to study the geometry of the Picard
functor Pic
X
/
S
when
X
/
S
is not necessarily cohomologically ²at. As an
application of this study, we are able to prove in full generality a theorem of
Gordon on the equivalence between the Artin-Tate and Birch-Swinnerton-
Dyer conjectures.
Recall that when
k
is algebraically closed, the special ±bers of elliptic
curves are classi±ed according to their Kodaira type, which is denoted by
a symbol
T
∈{
I
n
,
I
∗
n
,
n
∈
Z
≥
0
,
II
,
II
∗
,
III
,
III
∗
,
IV
,
IV
∗
}
. Given a type
T
and
a positive integer
m
, we denote by
mT
the new type obtained from
T
by
multiplying all the multiplicities of
T
by
m
.When
k
is algebraically closed,
the relationships between the type of a curve of genus 1 and the type of its
Jacobian can be summarized as follows.
Theorem 6.6.
Assume that k is algebraically closed. Let X
K
/
K be a smooth,
geometrically connected projective curve of genus
1
and let E
K
/
K be its
?
D.L. was supported by NSF under Grants No. 0070522 and 0302043.