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$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\F}{\Bbb F} \newcommand{\p}{\mathfrak{p}} $ Let $A$ be an abelian variety over a number field $F$. It is expected that the $L$-function of $A$ has analytic continuation to $\Bbb C$ and satisfies a functional equation relating $s$ to $2-s$. In that setting, the (generalized) Birch–Swinnerton-Dyer conjecture states that $$\mathrm{ord}_{s=1}(L(A_{/F},s)) = \mathrm{rk}_{\Z}(A(F)) =: r.$$

Originally, the conjecture for an elliptic curve $E$ over $\Q$ was $$\exists C>0,\quad \prod_{p \leq x} \dfrac{|E(\F_p)|}{p} \sim C \;\mathrm{log}(x)^r \qquad (x \to \infty).$$

My question is to know what is the analogue of the original conjecture, in the framework of abelian varieties over number fields.

My first guess would to replace to LHS by $$\prod_{N(\p) \leq x} L_{\p}(A_{/F}, N(\p)^{-1}),$$ where $L_{\p}(A_{/F},s)$ is the local factor of the L-function of $A$ at $\p$. But I'm not sure what the RHS should be. Typically, how does it depend on the dimension of $A$ or on the degree of the number field?

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$\newcommand{\p}{\mathfrak{p}}$By Theorem 6.3 of this paper by Keith Conrad, strong conjectures about $L(A,s)$ (stronger than GRH for this $L$-function, but still "believable"), imply that

$$ \prod_{N\p\le x}L_{\p}(A,N\p^{-1}) \sim C (\log x)^r $$ where $r$ is the order of vanishing at $s=1$, which by the usual BSD should be the rank of $A(K)$.

See also this question about the relation between the original and the modern formulation of BSD.

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  • $\begingroup$ Thank you for your answer. To apply theorem 5.11, I would need to check hypothesis (5.7), wouldn't I? How could I check it (assuming GRH, possibly...)? $\endgroup$ – Watson Sep 4 '18 at 11:43
  • $\begingroup$ @Watson Sorry, I quoted the wrong theorem! $\endgroup$ – Aurel Sep 4 '18 at 11:44
  • $\begingroup$ Moreover, do you know whether the constant $C$ depend on $A$ (in particular of $\mathrm{dim}(A)$)? $\endgroup$ – Watson Sep 4 '18 at 11:44
  • $\begingroup$ @Watson Konrad's paper also relates $C$ to the leading term, at $s=1$: this is Theorem 5.11. For the BSD formula for the leading term, see this question $\endgroup$ – Aurel Sep 4 '18 at 11:47
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    $\begingroup$ @Watson I want to point out that the mismatch between the coefficient in the (plausible but unproved!) asymptotic estimate for partial Euler products at the normalized central point $s=1/2$ and the leading Taylor coefficient of the $L$-function at that point is already appearing in the case of Hecke $L$-functions $s=1/2$: see Corollary 5.5 and Corollary 5.6. See Example 5.8 for some data in the case of one Dirichlet $L$-function. $\endgroup$ – KConrad Sep 4 '18 at 13:23

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