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Positivity of certain functions associated with analysis on elliptic surfaces. (English) Zbl 1237.11028

Let \(F\) be a number field and \(\mathcal{O}_F\) its ring of integers. Denote by \(E\) an elliptic curve over \(F\) and let \(\mathcal{E}\rightarrow \text{Spec}(\mathcal{O}_F)\) be a proper regular model of \(E\) which has normal crossing and the reduction in residual characteristic 2 and 3 is good or multiplicative.
The Hasse zeta function \(\zeta_{\mathcal{E}}(s)=\prod_{x\in \mathcal{E}_0}(1-|k(x)|^{-s})^{-1}\), where \(x\) runs through the closed points of \(\mathcal{E}\) and \(k(x)\) is the finite residue field of \(x\), satisfies the equality \[ \zeta_{\mathcal{E}}(s)=\eta_{\mathcal{E}}(s)\frac{\zeta_F(s)\zeta_F(s-1)}{L(E,s)} \] where \(L(E,s)\) is the Hasse-Weil \(L\)-function of \(E\), \(\zeta_F(s)\) the Dedekind zeta function and \(\eta_{\mathcal{E}}(s)\) is a meromorphic function defined from the singular fibers of \(\mathcal{E}\).
I. Fesenko proposed a new approach to study \(\zeta_{\mathcal{E}}(s)\) in terms of two dimensional analysis attached to the arithmetic elliptic surface [Doc. Math., J. DMV Extra Vol., 261–284 (2003; Zbl 1130.11335); J. K-Theory 5, No. 3, 437–557 (2010; Zbl 1225.14019); Proc. St. Petersbg. Math. Soc. 12. Transl. from the Russian. Providence, RI: American Mathematical Society (AMS). Transl.. Ser. 2. Am. Math. Soc. 219, 149–165 (2006; Zbl 1203.11080)]. Fesenko constructed a one variable function \(w_{\mathcal{E}}(s)\) that is a boundary term of the two dimensional zeta integral and which is related with the root number and conjecturally satisfies the functional equation \(w_{\mathcal{E}}(s)=w_{\mathcal{E}}(2-s)\). This zeta integral depends on a set \(S\), where \(S\) is the set of all fibers and a finitely many horizontal curves on \(\mathcal{E}\) (for example all fibers and the horizontal curves given by the zero section and the generators for \(E(F)\)). Fesenko worked under the assumption that the set \(S\) contains only one horizontal curve: the image of the zero section.
I. Fesenko proved the following result concerning the Riemann hypothesis [Mosc. Math. J. 8, No. 2, 273–317 (2008; Zbl 1158.14023)] [op. cit. (Zbl 1225.14019)]: Suppose \(L(E,s)\) has meromorphic continuation, satisfies the expected functional equation, \(L(E,s)\) has no real zeros in \((1,2)\) and satisfies the following assumption (F-1): there exists \(x_0>0\) such that the 4-th log derivative of a real-valued function \(h_{\mathcal{E}}(s)\), renamed \(Z_{\mathcal{E},S}(s)\), (where \(w_{\mathcal{E}}(s)=\int_0^1h_{\mathcal{E}}(x)x^{s-3}\,dx\)) does not change its sign in \((0,x_0)\). Then the Riemann hypothesis is true, i.e., all the poles in the critical strip of \[ \zeta_F(s/2)\frac{\zeta_F(s)\zeta_F(s-1)}{L(E,s)} \] lie in the line \(\text{Re}(s)=1\).
The paper under review studies how necessary (F-1) is to obtain the Riemann hypothesis.
M. Suzuki, in the paper under review, assumes that \(S\) contains \(I\) (\(<\infty\)) horizontal curves including the image of the zero section, in particular \(Z_{\mathcal{E},S}(s)\) is the \((2+2I)\)-th log derivative of \(h_{\mathcal{E}}(s)\), and first the author deals with some sort of generalizations of the above Fesenko results.
The main result of the paper is: Assume \(L(E,s)\) has analytic continuation, satisfies the expected functional equation and satisfies the Riemann hypothesis for \(L(E,s)\), then \[ |Z_{\mathcal{E},S}(x)|\leq C_{S,\mathcal{E},\varepsilon} x^{1-\varepsilon} \] for every \(\varepsilon>0\) and for some constant \(C_{S,\mathcal{E},\varepsilon}\); moreover, under additional hypotheses (\(\zeta_F(1/2)\neq 0\) and other reasonable assumptions), the author obtains a limit expression for \(Z_{\mathcal{E},S}(x)\) and then deduces that there exists \(x_{\mathcal{E}}>0\) such that \(Z_{\mathcal{E},S}(x)\) is positive for all \(x\in(0,x_{\mathcal{E}})\), obtaining in those cases that (F-1) is a necessary condition for the Riemann hypothesis.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11M41 Other Dirichlet series and zeta functions
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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References:

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