[Une traduction française suit la version anglaise.]


The question is only about elliptic curves $E$ over $\mathbb{Q}$ and concerns only the aspect 


(order of vanishing of $L(E,s)$ at $s=1$)$\ =\ $(rank of $E(\mathbb{Q})$).


Let $r$ be the LHS and $d$ the RHS, so that (a special case of ) the Birch and Swinnerton-Dyer Conjecture is


**BSD?.** $r=d$.


By the end of the last millenium, we knew


**Theorem** (1977--2000). *If* $\ r=0,1$, *then* $d=r$ (*and* $\ \operatorname{Sha}(E)$ *is finite*).


Some years ago, I heard that there was some progress in proving $(r>0)\Longrightarrow (d>0)$ under the assumption of the finiteness of $\operatorname{Sha}(E)$. What is the current status of the


**Statement.** *Suppose that* $\operatorname{Sha}(E)$ *is finite. If* $r>1$, *then* $d>0$ ?




---


*L'état actuel de la conjecture de Birch et Swinnerton-Dyer*


On s'interesse uniquement aux courbes abéliennes $A$ sur $\mathbf{Q}$ et à
l'aspect 


(ordre d'annulation de $L(A,s)$ en $s=1$)$\ =\ $(rang de $A(\mathbf{Q})$).


Désignons par $r$ le membre de gauche et par $d$ le membre de droite, de sorte
que la conjecture de Birch et Swinnerton-Dyer prédit (en particulier)


**BSD?** $r=d$.


Vers la fin du millénaire dernier, on avait démontré le


**Théorème** (1977--2000). *Si* $r=0,1$, *alors* $d=r$ (*et* $\ \operatorname{Cha}(A)$ *est fini*). 


Il y a quelques années, j'avais entendu dire qu'on a fait des progrès
concernant l'implication $(r>0)\Longrightarrow(d>0)$ sous l'hypothèse de la
finitude de Cha$(A)$. Quel est l'état actuel de l'


**Énoncé.** *Supposons que* $\operatorname{Cha}(A)$ *est fini*. *Si* $r>1$, *alors*
$d>0$ ?



The current status of the Birch & Swinnerton-Dyer Conjecture
The parity conjecture is known, i.e. it is known that if the order of vanishing of the $L$-function is even/odd, then the corank of $p$-Selmer is of the same parity (and I think this is known for every $p$ at this point; Nekovar handled the good ordinary or multiplicative case,
and B.D. Kim the good supersingular case. T. and V. Dokchitser then gave a new proof that handled the general case). This would imply that if Sha(E) is finite (even after passing to the $p$-part of Sha for some prime $p$) and the $L$-function has odd order vanishing, then $E({\mathbb Q})$ has positive rank.


There has also been recent work on establishing cases of positive even order vanishing, and trying to prove that the Selmer group has rank at least two. This has been investigated by both Bellaiche--Chenevier and Skinner--Urban. I don't know precise statements though, and I'm not sure if either pair of authors can handle elliptic curves. (In both cases, the arguments involve deforming along an eigenvariety, and there are problems at low weights. So they may only have results for modular forms of weight $k > 2$.)


Incidentally, although it wasn't part of your question, I don't think anything new is known about finiteness of Sha, beyond what you recalled in your question.


