Let
E
E
be an elliptic curve over
Q
\mathbb {Q}
and let
ϱ
\varrho
be an odd, irreducible two-dimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the Hasse-Weil-Artin
L
L
-series
L
(
E
,
ϱ
,
s
)
L(E,\varrho ,s)
, namely, the implication
\[
L
(
E
,
ϱ
,
1
)
≠
0
⇒
(
E
(
H
)
⊗
ϱ
)
G
a
l
(
H
/
Q
)
=
0
,
L(E,\varrho ,1) \ne 0\quad \Rightarrow \quad (E(H)\otimes \varrho )^{\mathrm {Gal}(H/\mathbb {Q})} = 0,
\]
where
H
H
is the finite extension of
Q
\mathbb {Q}
cut out by
ϱ
\varrho
. The proof relies on
p
p
-adic families of global Galois cohomology classes arising from Beilinson-Flach elements in a tower of products of modular curves.