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BirchSwinnertonDyer

Birch-Swinnerton-Dyer conjecture

The Birch-Swinnerton-Dyer conjecture (BSD) is a central conjecture in number theory concerning elliptic curves over the rational numbers and their L-functions. It was proposed by Bryan Birch and Peter Swinnerton-Dyer in the 1960s, following computational observations about the rank of the group of rational points and the behavior of the L-function at s = 1.

For an elliptic curve E over Q, its Hasse-Weil L-function L(E,s) is conjectured to extend to the

In its strong form, the leading coefficient of the Taylor expansion of L(E,s) at s = 1 is

Evidence for BSD is strongest in low analytic ranks. For many curves with L(E,1) ≠ 0 (rank 0)

complex
plane
and
satisfy
a
functional
equation.
The
conjecture
asserts
that
the
order
of
vanishing
of
L(E,s)
at
s
=
1
equals
the
rank
of
the
group
E(Q)
of
rational
points.
In
other
words,
the
analytic
rank
equals
the
algebraic
rank.
predicted
to
be
given
by
a
formula
involving
arithmetic
invariants
of
E,
including
the
regulator
of
E(Q),
the
(conjectured
finite)
Tate-Shafarevich
group
Sha(E/Q),
the
Tamagawa
numbers,
the
real
period
Ω_E,
and
the
size
of
the
torsion
subgroup
E(Q)_tors.
This
predicted
formula
is
known
as
the
BSD
formula
and
expresses
a
precise
link
between
analytic
data
and
arithmetic
data.
and
for
those
with
L′(E,1)
≠
0
(rank
1),
results
of
Gross–Zagier
and
Kolyvagin
show
the
conjecture
holds
in
those
cases,
including
finiteness
of
Sha.
Nevertheless,
the
conjecture
remains
open
in
general
and
has
broad
generalizations
to
abelian
varieties
over
number
fields.
It
is
one
of
the
Clay
Mathematics
Institute’s
Millennium
Prize
Problems.