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What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each with its own gamma factor)

In the end, this question is about understanding the BSD conjecture for abelian varieties over $\mathbf{Q}$ a bit better.

If $X$ is an abelian variety over $\mathbf{Q}$, is there a Birch and Swinnerton-Dyer conjecture for the zeta function of its Néron model? Is it related to the Birch and Swinnerton-Dyer conjecture for the Hasse-Weil $L$-function of $X$?

EXAMPLE: let $E$ be an elliptic curve over $\mathbf{Q}$, $\mathcal{E}$ its Néron model over $\text{Spec}(\mathbf{Z})$.

The zeta function of $\mathcal{E}$ is, by definition: $$Z(\mathcal{E}/\mathbf{Z},s) := \prod_{(p)\in\text{Spec}(\mathbf{Z})^0}Z(\mathcal{E}_p,p^{-s}),$$ where $\text{Spec}(\mathbf{Z})^0$ is the set of closed points of $\text{Spec}(\mathbf{Z})$, $\mathcal{E}_p$ is the mod $p$ fiber of $\mathcal{E}$, a variety over $\mathbf{F}_p$, and

$$Z(\mathcal{E}_p, T) = \frac{1 - a_pT + pT^2}{(1-T)(1-pT)},$$ while for the Hasse-Weil $L$-function $$L(E/\mathbf{Q},s) := \prod_{p\ {\rm good}}\frac{1}{1-a_pp^{-s} + p^{1-2s}}\cdot\prod_{p\ {\rm split\ mult.}}\frac{1}{1-p^{-s}}\cdot\prod_{p\ {\rm non-split\ mult.}}\frac{1}{1+p^{-s}}\cdot\prod_{p\ {\rm additive}}1.$$ where $a_p$ is the size of $\mathcal{E}_p(\mathbf{F}_p)$.

It seems: $$(*)\ \ \ Z(\mathcal{E}/\mathbf{Z},s) = \frac{\zeta(s)\zeta(s-1)}{L(E/\mathbf{Q},s)}.$$

We can introduce gamma factors and get

$$\hat{Z}(\mathcal{E}/\mathbf{Z},s) = \frac{\hat{\zeta}(s)\hat{\zeta}(s-1)}{\hat{L}(E/\mathbf{Q},s)}$$ where $\hat{\zeta}(s)$ is the completed Riemann zeta function, and $\hat{L}(E/\mathbf{Q},s)$ is the Hasse-Weil $L$-function of $E$ multiplied by its gamma factor as described by Serre.

PRECISE QUESTIONS:

(1.1): Is the BSD conjecture about $\hat{L}(E/\mathbf{Q},s)$? Ie. Is it true that $\text{ord}_{s = 1}\hat{L}(E/\mathbf{Q},s) = \text{ord}_{s=1}L(E/\mathbf{Q},s)$? (It looks the answer is yes because the gamma factor doesn't vanish at $s=1$). Is the BSD formula for $L$ and $\hat{L}$ the same? (ie. do we have $\text{Res}_{s=1}L(E/\mathbf{Q},s) = \text{Res}_{s=1}\hat{L}(E/\mathbf{Q},s)$?)

(1.2): As a baby example of (1.1). For $\zeta_K(s)$ the Dedekind zeta function of a number field $K$, and $\hat{\zeta}_K(s)$ the completed Dedekind zeta function, $\text{Res}_{s=1}\zeta_K(s)$ is predicted by the class number formula. What's the relation with $\text{Res}_{s=1}\hat{\zeta}_K(s)$? The former is $$\text{Res}_{s=1}\zeta_K(s) = \frac{2^{r_1}(2\pi)^{r_2} h_K R_K}{\omega_K\sqrt{|\Delta_K|}},$$ with usual meanings.

If we call $$\zeta_{K,\infty}(s) := |\Delta_K|^{s/2}\Gamma_{\mathbf{R}}(s)^{r_1}\Gamma_{\mathbf{C}}(s)^{r_2}$$ then the latter is $$\text{Res}_{s=1}\hat{\zeta}_K(s) = \zeta_{K,\infty}(1)\cdot\text{Res}_{s=1}\zeta_K(s) = 2^{r_1-r_2}\pi^{-r_1/2}\Gamma(1/2)^{r_1}\Gamma(1)^{r_2}\cdot \frac{ h_K R_K}{\omega_K}.$$

Does this expression mean anything? I was expecting to get a cleaner expression for the completed zeta function.

It looks the answer to (1.1), then, should be $$\text{Res}_{s=1}\hat{L}(E/\mathbf{Q},s) = L_{\infty}(1)\cdot \text{Res}_{s=1}L(E/\mathbf{Q},s).$$

(2): (the question I am most interested in)

Is there a BSD conjecture for $\hat{Z}(\mathcal{E}/\mathbf{Z},s)$ (call it BSD$(\hat{Z})$)? Is it related to the BSD conjecture for $\hat{L}(E/\mathbf{Q},s)$ (call it BSD$(L)$)?

(specifically, do we have BSD$(\hat{Z})\Rightarrow$ BSD$(L)$?)

(3): What is a formula analogous to $(*)$, for higher dimensional abelian varieties and their zeta vs Hasse-Weil $L$-functions?

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EDIT: an equivalent question I would be very interested to know the answer of.

In the function field case and for elliptic curves, Tate proved the BSD conjecture (for each elliptic curve over a function field) is equivalent to the Artin-Tate conjecture for a canonical elliptic surface (over a finite field). I think this has been generalized to abelian varieties. Is it possible to prove something analogous over number fields? ie. is it possible to deduce the BSD conjecture for the Hasse-Weil $L$-function of an abelian variety from a number field analog of the Artin-Tate conjecture (where probably the leading coefficient formula involves torsion and volumes on higher Chow groups) for the zeta function of an appropriate arithmetic scheme?

(without making use of Weil-êtale cohomologies, etc)

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    $\begingroup$ There is a conjecture of Lichtenbaum for the order of vanishing and leading term of zeta functions of arithmetic schemes at $s=0$. In the case of smooth projective schemes over $\mathbf{Z}$, the conjecture about the leading term is equivalent to the Tamagawa number conjecture by Bloch--Kato, as shown by B. Morin in the article "Zeta functions of regular arithmetic schemes at $s=0$". $\endgroup$ – François Brunault Dec 21 '17 at 8:10
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    $\begingroup$ You should probably restrict to arithmetic schemes which are smooth and projective over the ring of integers of a number field to get good properties of the zeta function. In particular, you may want to replace the Néron model of an elliptic curve by the minimal proper regular model. $\endgroup$ – François Brunault Dec 21 '17 at 8:12
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    $\begingroup$ First, when changing the local factor at some place (finite or infinite) should not be difficult to reformulate the BSD conjecture as the factors are explicit. For the function field, that is what Tate did to prove BSD up to the finiteness of Sha. $\endgroup$ – Chris Wuthrich Dec 21 '17 at 15:54
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    $\begingroup$ The minimal regular model or the Neron model look like good candidates from a geometric point of view, but the L-function is nicer when looking from a Galois representation point of view. As the previous comments point out, one has to be very careful with the local factors at bad primes. Your (*) is wrong. If the group of components is non-trivial then you would get multiple factors. If I remember well, the locally minimal Weierstrass equation is actually (and maybe strangely) better. $\endgroup$ – Chris Wuthrich Dec 21 '17 at 16:04
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    $\begingroup$ This is fairly elementary even beyond the case of curves. What do you mean, then? Finiteness of Brauer groups implies finiteness of Sha of the Néron model $\mathcal{A}^{\vee}$ of (the dual of) an abelian variety, but this doesn't mean there's a BSD formula for the zeta function of $\mathcal{A}$ that implies the BSD formula for the generic fiber of $\mathcal{A}$. Only over function fields we know finiteness of Sha proves the whole thing. $\endgroup$ – user113393 Dec 22 '17 at 6:29

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