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Let $E$ be an elliptic curve over $\mathbb{Q}$. As proved by Wiles et al., its $L$-series $L(E, s)$ is entire. Set $r := \mathrm{ord}_{s = 1} L(E, s)$, a value conjecturally equal to $\mathrm{dim}_{\mathbb{Q}} (E(\mathbb{Q}) \otimes_{\mathbb{Z}} \mathbb{Q})$. There is a $c \in \mathbb{C}^\times$ such that $L(E, s) \sim c \cdot (s - 1)^r$ as $s \rightarrow 1$. Birch and Swinnerton-Dyer predict a great deal about $c$, in particular, that

  1. $c \in \mathbb{R}$,
  2. $c \in \mathbb{R}_{> 0}$,
  3. $c/\Omega_E \in \mathbb{Q}$ if $r = 0$, where $\Omega_E$ is the volume of $E(\mathbb{R})$ computed with respect to "the" Neron differential of $E$, and
  4. $c/\Omega_E \in \mathbb{Q}_{> 0}$ if $r = 0$.

Which of these predictions are known to hold?

Partial answers and answers in special cases (e.g., in the case when $E$ has potential complex multiplication) are very welcome!

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    $\begingroup$ 1 and 2 are known. 3 and 4 are not right unless $r=0$, because the height pairing doesn't take rational values. If $r=0$, then maybe 3 and 4 are known by Kolyvagin, but I am not sure. $\endgroup$ – Felipe Voloch Jul 3 '14 at 2:13
  • $\begingroup$ Thanks for your comment! Could you give a reference where 1. and 2. are discussed (or a sketch how the argument goes)? You are right that 3. and 4. are incorrect as stated; I will add the $r = 0$ condition to them. $\endgroup$ – Question Mark Jul 3 '14 at 2:34
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    $\begingroup$ The identity $L(E,\bar{s})=\overline{L(E,s)}$, which is clear for $\Re(s)>3/2$, analytically continues. $\endgroup$ – Felipe Voloch Jul 3 '14 at 3:00
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    $\begingroup$ I think #2 in the $r=0$ case is Waldspurger in essence, and then a long series of papers concluding with Guo, will track reference (cf bottom pg 722 of Iwaniec/Sarnak perspectives paper). "On the positivity of the central value of automorphic $L$-functions for $GL(2)$", Duke Math 83 1996, 1-18. As IwSa says, Guo uses the relative trace formula, rather than $\Theta$-series. projecteuclid.org/euclid.dmj/1077244251 $\endgroup$ – NAME_IN_CAPS Jul 3 '14 at 3:32
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    $\begingroup$ I don't think Kolyvagin can say anything about the sign of the L-value. His argument is prime-by-prime and for most primes the formula of BSD then holds. But that will always miss the sign it seems to me. $\endgroup$ – Chris Wuthrich Jul 3 '14 at 9:11
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Let me summarise the comments above that give a full answer (correct me if I am wrong).

  1. The analytic continuation of $L(E,\bar{s})=\overline{L(E,s)}$ shows that $c\in\mathbb{R}$.

  2. If $r=0$, the fact that $c>0$ is proven in On the positivity of the central value of automorphic L-functions for GL(2) Duke Math 83 1996, 1-18. It would follow from the generalised Riemann hypothesis, too. For $r=1$, it follows from Gross-Zagier and the case $r=0$. I don't know about $r>1$.

  3. That $L(E,1)/\Omega_E$ is a rational number is a consequence of the theorem of Manin-Drinfeld on modular symbols.

  4. Is just 2+3. Note that $\Omega_E$ is defined to be positive, as it is the least positive real period of a Néron differential on $E$ (or twice that depending on your normalisation).

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