Talk:Zeta-function

$\zeta$-function

Zeta-functions in number theory are functions belonging to a class of analytic functions of a complex variable, comprising Riemann's zeta-function, its generalizations and analogues. Zeta-functions and their generalizations in the form of $L$-functions (cf. Dirichlet $L$-function) form the basis of modern analytic number theory. In addition to Riemann's zeta-function one also distinguishes the generalized zeta-function $\zeta(s,a)$, the Dedekind zeta-function, the congruence zeta-function, etc.

Riemann's zeta-function is defined by the Dirichlet series

\begin{equation}\label{sum} \zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},\quad s=\sigma+it,\end{equation}

which converges absolutely and uniformly in any bounded domain of the complex $s$-plane for which $\sigma\geq1+\delta$, $\delta>0$. If $\sigma>1$, a valid representation is the Euler product

\begin{equation}\label{prod} \zeta(s)=\prod_p\left(1-\frac{1}{p^s}\right)^{-1},\end{equation}

where $p$ runs through all prime numbers.

The identity of the series \ref{sum} and the product \ref{prod} is one of the fundamental properties of $\zeta(s)$. It makes it possible to obtain numerous relations connecting $\zeta(s)$ with important number-theoretic functions. E.g., if $\sigma>1$,

$$ \ln \zeta(s)=s\int_2^\infty\frac{\pi(x)}{x(x^s-1)}\,\mathrm{d}x,$$

$$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^s},$$

$$\frac{1}{\zeta(s)}=\sum_{n=1}^\infty\frac{\mu(n)}{n^s},\quad \zeta^2(s)=\sum_{n=1}^\infty\frac{\tau(n)}{n^s},$$

$$\frac{\zeta^2(s)}{\zeta(2s)}=\sum_{n=1}^\infty\frac{2^{\nu(n)}}{n^s},\quad\frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^\infty\frac{\lambda(n)}{n^s}.$$

Here $\pi(x)$ is the number of primes $\leq x$, $\Lambda(n)$ is the (von) Mangoldt function, $\mu(n)$ is the Möbius function, $\tau(n)$ is the number divisors of the number $n$, $\nu(n)$ is the number of different prime factors of $n$, and $\lambda(n)$ is the Liouville function. This accounts for the important role played by $\zeta(s)$ in number theory. As a function of a real variable, $\zeta(s)$ was introduced in 1737 by L. Euler [1], who proved that it could be expanded into the product \ref{prod}. The function was subsequently studied by P.G.L. Dirichlet and also, with extraordinary success, by P.L. Chebyshev [2] in the context of the problem of the distribution of prime numbers. However, the most deeply intrinsic properties of $\zeta(s)$ were discovered later, as a result of studying it as a function of a complex variable. This was first accomplished in 1876 by B. Riemann [3], who demonstrated the following assertions.

a) $\zeta(s)$ permits analytic continuation to the whole complex $s$-plane, in the form

\begin{equation}\label{cont} \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\frac{1}{s(s-1)}+\int_1^\infty\left( x^{-(1-s/2)}+x^{-(1-(1-s)/2)}\right)\theta(x)\,\mathrm{d}x,\end{equation}

where $\Gamma(\omega)$ is the gamma-function and

$$\theta(x)=\sum_{n=1}^\infty \exp(-\pi n^2x).$$

b) $\zeta(s)$ is a regular function for all values of $s$ except for $s=1$, where it has a simple pole with residue one, and it satisfies the functional equation

\begin{equation}\label{func}\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-(1-s)/2}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s).\end{equation}

This equation is known as Riemann's functional equation. For the function

$$ \xi(s)=\frac{s(s-1)}{2}\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s),$$

introduced by Riemann for studying the zeta-function and now known as Riemann's $\xi$-function, this equation assumes the form

$$ \xi(s)=\xi(1-s),$$

while if one puts

$$\Xi(t)=\xi\left(\frac{1}{2}+it\right),$$

it assumes the form

$$\Xi(t)=\Xi(-t).$$

This last function $\Xi$ is distinguished by the fact that it is an even entire function which is real for real $t$, and its zeros on the real axis correspond to the zeros of $\zeta(s)$ on the straight line $\sigma=1/2$.

c) Since $\zeta(s)\neq0$ for $\sigma>1$, by \ref{func} this function has only simple zeros at the points $s=-2\nu$, $\nu=1,2,\ldots,$ in the half-plane $\sigma<0$. These zeros are known as the trivial zeros of $\zeta(s)$. Also, $\zeta(s)\neq0$ for $0<s<1$. Thus, all non-trivial zeros of $\zeta(s)$ are complex numbers, lying symmetric with respect to both the real axis $t=0$ and the vertical line $\sigma=1/2$ and situated inside the strip $0\leq\sigma\leq1$. This strip is known as the critical strip.

Riemann also stated the following hypotheses.

1) The number $N(T)$ of zeros of $\zeta(s)$ in the rectangle $0\leq\sigma\leq1$, $0<t<T$ can be expressed by the formula

$$N(T)=\frac{1}{2\pi}T\ln T-\frac{1+\ln 2\pi}{2\pi}T+O(\ln T).$$

2) Let $\rho$ run through the non-trivial zeros of $\zeta(s)$. Then the series $\sum\lvert\rho\rvert^{-2}$ is convergent, while the series $\sum\lvert\rho\rvert^{-1}$ is divergent.

3) The function $\xi(s)$ can be represented in the form

$$ ae^{bs}\prod_\rho \left(1-\frac{s}{\rho}\right)e^{s/\rho}.$$

4) Let

$$ P(x)=\sum_{n\leq x}\frac{\Lambda(n)}{\ln n},$$

$$ P_0(x)=\frac{1}{2}[P(x+0)+P(x-0)].$$

Then, for $x\geq1$,

$$ P_0(x)=\mathrm{li} x-\sum_\rho\mathrm{li}x^\rho+\int_x^\infty\frac{\mathrm{d}u}{(u^2-1)\ln u}-\ln 2,$$

where $\mathrm{li} x$ is the integral logarithm:

$$\mathrm{li} e^w=\int_{-\infty+iv}^{u+iv}\frac{e^z}{z}\,\mathrm{d}z,\quad w=u+iv,\quad v<0\text{ or }v>0.$$

5) All non-trivial zeros of $\zeta(s)$ lie on the straight line $\sigma=1/2$.

Subsequent to Riemann, the problem on the value distribution and, in particular, the zero distribution of the zeta-function became very widely known and was studied by a large number of workers. Riemann's hypotheses 2 and 3 were proved by J. Hadamard in 1893, and it was proved that, in hypothesis 3, $a=1/2$ and $b=\ln 2+(1/2)\ln\pi-1-C/2$, where $C$ is the Euler constant; hypotheses 1 and 4 were established in 1894 by H. von Mangoldt, who also obtained the following important analogue of (5) for prime numbers. If

$$\Psi(x)=\sum_{n\leq x}\Lambda(n),\quad \Psi_0(x)=\frac{1}{2}[\Psi(x+0)-\Psi(x-0)],$$

then, for $x\geq1$,

$$ \Psi_0(x)=x-\sum_\rho\frac{x^\rho}{\rho}-\frac{\zeta'(0)}{\zeta(0)}-\frac{1}{2}\ln\left(1-\frac{1}{x^2}\right),$$

where $\rho=\beta+i\gamma$ runs through the non-trivial zeros of $\zeta(s)$, while the symbol $\sum_\rho x^\rho/\rho$ denotes the limit of the sum $\sum_{\lvert \gamma\rvert\leq T}x^\rho/\rho$ as $T\to\infty$. This formula shows, similarly to formula (5), that the problem of the distribution of primes in the natural number series is closely connected with the location of the non-trivial zeros of the function $\zeta(s)$.

The last hypothesis (hypothesis 5) has not yet (1993) been proved or verified. This is the famous Riemann hypothesis on the zeros of the zeta-function.

The function $\zeta(s)$ is unambiguously defined by its functional equation. More exactly, any function which can be represented by an ordinary Dirichlet series and which satisfies equation (4) coincides, under fairly broad conditions with respect to its regularity, with $\zeta(s)$, up to a constant factor [4].

If

$$ \chi(s)=\pi^{s-1/2}\frac{\Gamma(1-s/2)}{\Gamma(s/2)}$$

and $h>0$ is constant, the approximate functional equation

\begin{equation}\label{approx} \zeta(s)=\sum_{n\leq x}\frac{1}{n^s}+\chi(s)\sum_{n\leq y}\frac{1}{n^{1-s}}+O(x^{-\sigma})+O(\lvert t\rvert^{1/2-\sigma}y^{\sigma-1}),\end{equation}

obtained in 1920 by G.H. Hardy and J.E. Littlewood [4], is valid for $0<\sigma<1$, $x>h$, $y>h$, $2\pi xy=\lvert t\rvert$. This equation is important in the modern theory of the zeta-function and its applications. There exist general methods by which such results may be obtained not only for the class of zeta-functions, but in general for Dirichlet functions with a Riemann-type functional equation \ref{func}. The most complete result in this direction has been shown in [5]; in the case of $\zeta(s)$ it leads, for any $\tau$ with $\lvert \arg \tau\rvert<\pi/2$, to the relation

$$\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-s/2}\sum_{n=1}^\infty\frac{\Gamma(s/2,\pi n^2\tau)}{n^s}+\pi^{-(1-s)/2}\sum_{n=1}^\infty\frac{\Gamma((1-s)/2,\pi n^2/\tau)}{n^{1-s}}-\frac{\tau^{(s-1)/2}}{1-s}-\frac{\tau^{s/2}}{s},$$

where $\Gamma(z,x)$ is the incomplete gamma-function. For

$$\tau=\Delta^2\exp\left[ i\left(\frac{\pi}{2}-\frac{1}{\lvert t\rvert}\right)\mathrm{sign} t\right],\quad \Delta>0,$$

one obtains the approximate equation \ref{approx}; for $\tau=1$ this relation becomes identical with the initial formula \ref{func}.

The principal problem in the theory of the zeta-function is the problem of the location of its non-trivial zeros and, in general, of its values within the range $1/2\leq \sigma\leq 1$. The main directions of research conducted on the zeta-function include: the determination of the widest possible domain to the left of the straight line $\sigma=1$ where $\zeta(s)\neq0$; the problem of the order and of the average values of the zeta-function in the critical strip; estimates of the number of zeros of the zeta-function on the straight line $\sigma=1/2$ and outside it, etc.

The first non-trivial result on the boundary for the zeros of the zeta-function was obtained in 1896 by Ch.J. de la Vallée-Poussin, who showed that there exists a constant $A>0$ such that

\begin{equation}\label{zerofree}\zeta(s)\neq0\qquad\text{ if }\sigma\geq1-\frac{A}{\ln^\alpha(\lvert t\rvert+2)}\text{ with }\alpha\geq1.\end{equation}

Other related approximations are connected with the approximate equation \ref{approx} and with the development of methods for estimating trigonometric sums.

The most powerful method for making estimates of this kind must be credited to I.M. Vinogradov (cf. Vinogradov method). The latest (to 1978) bound on the boundary of the zero-free domain for the zeta-function was obtained by Vinogradov in 1958 [7]. It is of the form \ref{zerofree} with $\alpha>2/3$. The formula

$$\pi(x)=\mathrm{li}x+O\left(xe^{-B\ln^{3/5}x}\right)$$

is the corresponding statement for prime numbers. There exists a certain connection between the growth of the modulus of the function $\zeta(s)$ and the absence of zeros in a neighbourhood of the straight line $\sigma=1$. Thus, \ref{zerofree} with $\alpha>2/3$ is the result of the estimates

$$ \zeta(1+it)=O\left(\ln^{2/3}\lvert t\rvert\right),\qquad\frac{1}{\zeta(1+it)}=O\left(\ln^{2/3}\lvert t\rvert\right),\quad \lvert t\rvert>2.$$

It is known, on the other hand [4], that

$$ \overline{\lim}_{t\to \infty}\frac{\lvert \zeta(1+it)\rvert}{\ln\ln t}\geq e^C,\quad \overline{\lim}_{t\to\infty}\frac{\lvert \zeta(1+it)\rvert^{-1}}{\ln\ln t}\geq\frac{6}{\pi^2}e^C,$$

and, if Riemann's hypothesis is valid, these bounds should not exceed $2e^C$ and $(12/\pi^2)e^C$, respectively.

The order of the zeta-function in the critical strip is the greatest lower bound $\eta(\sigma)$ of the numbers $\nu$ such that $\zeta(\sigma+it)=O(\lvert t\rvert^\nu)$. If $$\sigma>1$, $\eta(\sigma)=0$, and if $\sigma<0$, then $\eta(\sigma)=(1/2)-\sigma$. The exact values of the function $\eta(\sigma)$ for $0\leq\sigma\leq 1$ are unknown. According to the simplest assumption (the [[Lindelöf hypothesis|Lindelöf hypothesis]]) $$ \eta(\sigma)=\frac{1}{2}-\sigma\text{ if }\sigma<\frac{1}{2}\quad\text{ and }\quad\eta(\sigma)=0\text{ if }\sigma>\frac{1}{2}.$$ This is the equivalent to the statement that $$ \zeta\left(\frac{1}{2}+it\right)=O(\lvert t\rvert^{\epsilon})\quad\text{ for any }\epsilon>0.$$ If $$\sigma>1/2$, the estimate '"`UNIQ-MathJax28-QINU`"'\zeta\left(\frac{1}{2}+it\right)=O(\lvert t\rvert^{\epsilon+15/32})'"`UNIQ-MathJax29-QINU`"'\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2k}\,\mathrm{d}t'"`UNIQ-MathJax30-QINU`"'\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2}\,\mathrm{d}t=\ln T+2C-1-\ln 2\pi+O\left(\frac{\ln T}{\sqrt{T}}\right),'"`UNIQ-MathJax31-QINU`"'\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{4}\,\mathrm{d}t=\frac{\ln^4T}{2\pi^2}+O(\ln^3T).'"`UNIQ-MathJax32-QINU`"'\lim_{T\to\infty}\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2}\,\mathrm{d}t=\zeta(2\sigma)'"`UNIQ-MathJax33-QINU`"'\lim_{T\to\infty}\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{4}\,\mathrm{d}t=\frac{\zeta^4(2\sigma)}{\zeta(4\sigma)}'"`UNIQ-MathJax34-QINU`"'\lim_{T\to\infty}\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2k}\,\mathrm{d}t=\sum_{n=1}^\infty\frac{\tau_k^2(n)}{n^{2\sigma}},'"`UNIQ-MathJax35-QINU`"'\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2k}\,\mathrm{d}t\sim \sum_{n=1}^\infty\frac{\tau_k^2(n)}{n^{2\sigma}}'"`UNIQ-MathJax36-QINU`"'z099/z099260/z099260168.png"'"`UNIQ-MathJax37-QINU`"'z099/z099260/z099260171.png"'"`UNIQ-MathJax38-QINU`"'z099/z099260/z099260176.png"'"`UNIQ-MathJax39-QINU`"'z099/z099260/z099260179.png"'"`UNIQ-MathJax40-QINU`"'z099/z099260/z099260181.png"'"`UNIQ-MathJax41-QINU`"'z099/z099260/z099260182.png"'"`UNIQ-MathJax42-QINU`"'z099/z099260/z099260184.png"'"`UNIQ-MathJax43-QINU`"'z099/z099260/z099260188.png"'"`UNIQ-MathJax44-QINU`"'z099/z099260/z099260192.png"'"`UNIQ-MathJax45-QINU`"'z099/z099260/z099260194.png"'"`UNIQ-MathJax46-QINU`"'z099/z099260/z099260204.png"'"`UNIQ-MathJax47-QINU`"'z099/z099260/z099260206.png"'"`UNIQ-MathJax48-QINU`"'z099/z099260/z099260208.png"'"`UNIQ-MathJax49-QINU`"'z099/z099260/z099260228.png"'"`UNIQ-MathJax50-QINU`"'z099/z099260/z099260237.png"'"`UNIQ-MathJax51-QINU`"'z099/z099260/z099260246.png"'"`UNIQ-MathJax52-QINU`"'z099/z099260/z099260251.png"'"`UNIQ-MathJax53-QINU`"'z099/z099260/z099260253.png"'"`UNIQ-MathJax54-QINU`"'z099/z099260/z099260254.png"'"`UNIQ-MathJax55-QINU`"'z099/z099260/z099260279.png"'"`UNIQ-MathJax56-QINU`"'z099/z099260/z099260293.png"'"`UNIQ-MathJax57-QINU`"'z099/z099260/z099260314.png"'"`UNIQ-MathJax58-QINU`"'z099/z099260/z099260320.png"'"`UNIQ-MathJax59-QINU`"'z099/z099260/z099260336.png"'"`UNIQ-MathJax60-QINU`"'z099/z099260/z099260347.png"'"`UNIQ-MathJax61-QINU`"'z099/z099260/z099260352.png"'"`UNIQ-MathJax62-QINU`"'z099/z099260/z099260357.png"'"`UNIQ-MathJax63-QINU`"'z099/z099260/z099260363.png"'"`UNIQ-MathJax64-QINU`"'z099/z099260/z099260370.png"'"`UNIQ-MathJax65-QINU`"'z099/z099260/z099260381.png"'"`UNIQ-MathJax66-QINU`"'z099/z099260/z099260397.png"'"`UNIQ-MathJax67-QINU`"'z099/z099260/z099260405.png"$$ where $z099/z099260/z099260406.png"$ is the assumed finite order of the Shafarevich–Tate group of the locally trivial [[Principal homogeneous space|principal homogeneous space]] of the variety $z099/z099260/z099260407.png"$, $z099/z099260/z099260408.png"$ is the determinant of the bilinear form on the group of rational points of the variety $z099/z099260/z099260409.png"$, which is obtained from the height (cf. [[Height, in Diophantine geometry|Height, in Diophantine geometry]]) of points, and $z099/z099260/z099260410.png"$ and $z099/z099260/z099260411.png"$ are the orders of the torsion subgroups in the group of rational points on $z099/z099260/z099260412.png"$ and the dual Abelian variety. This expression generalizes the expression for the residue of the Dedekind zeta-function at the point $z099/z099260/z099260413.png"$ which is familiar in algebraic number theory. One difficulty involved in demonstrating the Birch–Swinnerton-Dyer conjecture is the fact that group $z099/z099260/z099260414.png"$ has not yet (1978) been fully computed for any curve. The analogue of the hypothesis has been proved for curves defined over a field of functions, but even in this case it had been necessary to assume the finiteness of the [[Brauer group|Brauer group]], which here plays the role of the group $z099/z099260/z099260415.png"$ [[#References|[5]]]. In his study of the action of the Galois group on algebraic cycles of varieties, Tate [[#References|[13]]] proposed a conjecture on the poles of the functions $z099/z099260/z099260416.png"$ for even values of $z099/z099260/z099260417.png"$, to wit, that the function $z099/z099260/z099260418.png"$ has, at the point $z099/z099260/z099260419.png"$, a pole of order equal to the rank of the group of algebraic cycles of codimension $z099/z099260/z099260420.png"$. This statement is closely connected with Tate's conjecture on algebraic cycles. For the various approaches leading to proofs of these conjectures, and for various arguments in favour of them, see [[#References|[5]]], [[#References|[7]]], [[#References|[12]]], [[#References|[13]]], [[#References|[17]]]. Quite apart from the concept of the zeta-function just described, zeta-functions which are Mellin transforms of modular forms have been studied in the theory of algebraic groups and automorphic functions. Weil noted in 1967 that a consequence of the general hypotheses on the function $z099/z099260/z099260421.png"$ for an elliptic curve $z099/z099260/z099260422.png"$ over $z099/z099260/z099260423.png"$ is that the curve $z099/z099260/z099260424.png"$ is uniformized by modular functions, while the function $z099/z099260/z099260425.png"$ is the Mellin transform of the modular form corresponding to a differential of the first kind on $z099/z099260/z099260426.png"$. This observation led to the assumption that the functions $z099/z099260/z099260427.png"$ of any scheme $z099/z099260/z099260428.png"$ are Mellin transforms of the respective modular forms. Basic results on this problem were obtained by E. Jacquet and R. Langlands [[#References|[7]]], [[#References|[9]]]. In particular, they constructed an extensive class of Dirichlet series satisfying a certain functional equation and expandable into an Euler product which may be represented as the Mellin transform of modular forms on the group $z099/z099260/z099260429.png"$. Meeting the requirements of this theorem is directly related to the conjectures on the general properties of zeta-functions discussed above. Their verification is as yet possible only for curves defined over a field of functions. From 1970 on, the studies of $z099/z099260/z099260430.png"$-adic zeta-functions of algebraic number fields [[#References|[14]]] stimulated a similar approach to the zeta-functions of schemes — mainly elliptic curves. The problems involved, which greatly resemble those discussed above, are reviewed in [[#References|[9]]]. The zeta-function of an elliptic curve over $z099/z099260/z099260431.png"$ is closely connected with the one-dimensional [[Formal group|formal group]] of the curve, and they completely define each other [[#References|[16]]]. ===='"`UNIQ--h-5--QINU`"'References==== <table><tr><td valign="top">[1]</td> <td valign="top"> E. Artin, "Quadratische Körper im Gebiet der höheren Kongruenzen I, II" ''Math. Z.'' , '''19''' (1924) pp. 153–246 </td></tr><tr><td valign="top">[2]</td> <td valign="top"> A. Weil, "Courbes algébriques et variétés abéliennes. Sur les courbes algébriques et les varietés qui s'en deduisent" , Hermann (1948) </td></tr><tr><td valign="top">[3]</td> <td valign="top"> A. Weil, "Numbers of solutions of equations in finite fields" ''Bull. Amer. Math. Soc.'' , '''55''' : 5 (1949) pp. 497–508 [https://mathscinet.ams.org/mathscinet/article?mr=0029393 MR0029393] [https://zbmath.org/?q=an%3A0032.39402 Zbl 0032.39402] </td></tr><tr><td valign="top">[4]</td> <td valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–307 [https://mathscinet.ams.org/mathscinet/article?mr=0340258 MR0340258] [https://zbmath.org/?q=an%3A0314.14007 Zbl 0314.14007] [https://zbmath.org/?q=an%3A0287.14001 Zbl 0287.14001] </td></tr><tr><td valign="top">[5]</td> <td valign="top"> A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , ''Dix exposés sur la cohomologie des schémas'' , North-Holland & Masson (1968) </td></tr><tr><td valign="top">[6]</td> <td valign="top"> B. Dwork, "A deformation theory for the zeta-function of a hypersurface" , ''Proc. Internat. Congress Mathematicians (Djursholm, 1963)'' , Almqvist & Weksell (1963) pp. 247–259 [https://mathscinet.ams.org/mathscinet/article?mr=0175895 MR0175895] [https://zbmath.org/?q=an%3A0196.53302 Zbl 0196.53302] </td></tr><tr><td valign="top">[7]</td> <td valign="top"> E. Jacquet, "Automorphic forms on $z099/z099260/z099260432.png"$" , '''1''' , Springer (1970) [https://mathscinet.ams.org/mathscinet/article?mr=0401654 MR0401654] [https://zbmath.org/?q=an%3A0236.12010 Zbl 0236.12010] </td></tr><tr><td valign="top">[8]</td> <td valign="top"> Yu.I. Manin, "Cyclotomic fields and modular curves" ''Russian Math. Surveys'' , '''26''' : 6 (1971) pp. 7–78 ''Uspekhi Mat. Nauk'' , '''26''' : 6 (1971) pp. 7–71 [https://mathscinet.ams.org/mathscinet/article?mr=0401653 MR0401653] [https://zbmath.org/?q=an%3A0266.14012 Zbl 0266.14012] </td></tr><tr><td valign="top">[9]</td> <td valign="top"> J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , ''Modular functions of one variable. 2–3'' , ''Lect. notes in math.'' , '''349; 350''' , Springer ( 1973) [https://mathscinet.ams.org/mathscinet/article?mr=0323724 MR0323724] </td></tr><tr><td valign="top">[10]</td> <td valign="top"> J.-P. Serre, "Zeta and $z099/z099260/z099260433.png"$-functions" O.F.G. Schilling (ed.) , ''Arithmetical Algebraic geometry (Proc. Purdue Conf. 1963)'' , Harper & Row (1965) pp. 82–92 [https://mathscinet.ams.org/mathscinet/article?mr=0194396 MR0194396] [https://mathscinet.ams.org/mathscinet/article?mr=0190106 MR0190106] </td></tr><tr><td valign="top">[11]</td> <td valign="top"> J.-P. Serre, "Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures)" ''Sem. Delange–Pisot–Poitou'' , '''19''' (1969/70)</td></tr><tr><td valign="top">[12]</td> <td valign="top"> P. Swinnerton-Dyer, "The conjectures of Birch and Swinnerton-Dyer and of Tate" T.A. Springer (ed.) , ''Local Fields (Proc. Conf. Driebergen, 1966)'' , Springer (1967) pp. 132–157 [https://zbmath.org/?q=an%3A0197.47101 Zbl 0197.47101] </td></tr><tr><td valign="top">[13]</td> <td valign="top"> J.T. Tate, "Algebraic cycles and poles of zeta-functions" O.F.G. Schilling (ed.) , ''Arithmetical Algebraic geometry (Proc. Purdue Conf. 1963)'' , Harper & Row (1965) pp. 93–110 [https://mathscinet.ams.org/mathscinet/article?mr=0225778 MR0225778] [https://zbmath.org/?q=an%3A0213.22804 Zbl 0213.22804] </td></tr><tr><td valign="top">[14]</td> <td valign="top"> I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian) </td></tr><tr><td valign="top">[15]</td> <td valign="top"> G. Shimura, "Introduction to the mathematical theory of automorphic functions" , Princeton Univ. Press (1971) </td></tr><tr><td valign="top">[16]</td> <td valign="top"> T. Honda, "Formal groups and zeta-functions" ''Osaka J. Math.'' , '''5''' (1968) pp. 199–213 [https://mathscinet.ams.org/mathscinet/article?mr=0249438 MR0249438] [https://zbmath.org/?q=an%3A0169.37601 Zbl 0169.37601] </td></tr><tr><td valign="top">[17]</td> <td valign="top"> A.N. Parshin, "Arithmetic on algebraic varieties" ''J. Soviet Math.'' , '''1''' : 5 (1973) pp. 594–620 ''Itogi Nauk. Algebra. Topol. Geom. 1970'' (1970/71) pp. 111–151 [https://zbmath.org/?q=an%3A0284.14004 Zbl 0284.14004] </td></tr></table> ''A.N. Parshin'' ===='"`UNIQ--h-6--QINU`"'Comments==== The conjectures of Birch and Swinnerton-Dyer have been generalized by S. Bloch and P. Beilinson to conjectures relating the ranks of Chow groups obtained from algebraic cycles with orders of poles of zeta-functions. See [[#References|[a6]]]–[[#References|[a8]]]. The Tate–Shafarevich group of certain elliptic curves over number fields has recently been computed (–[[#References|[a5]]]). As predicted, it is finite in these cases. The Weil conjectures and their proofs have been extended to the case of arbitrary schemes of finite type [[#References|[a1]]]. ===='"`UNIQ--h-7--QINU`"'References==== <table><tr><td valign="top">[a1]</td> <td valign="top"> P. Deligne, "La conjecture de Weil, II" ''Publ. Math. IHES'' , '''52''' (1980) pp. 137–252 [https://mathscinet.ams.org/mathscinet/article?mr=0601520 MR0601520] [https://zbmath.org/?q=an%3A0456.14014 Zbl 0456.14014] </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> E. Freitag, R. Kiehl, "Étale cohomology and the Weil conjecture" , Springer (1988) [https://mathscinet.ams.org/mathscinet/article?mr=0926276 MR0926276] [https://zbmath.org/?q=an%3A0643.14012 Zbl 0643.14012] </td></tr><tr><td valign="top">[a3a]</td> <td valign="top"> V. Kolyvagin, "Finiteness of $z099/z099260/z099260434.png"$ and $z099/z099260/z099260435.png"$ for a subclass of Weil curves" ''Math. USSR Izv.'' , '''33''' (1989) ''Izv. Akad. Nauk SSSR'' , '''52''' (1988) pp. 522–540 [https://mathscinet.ams.org/mathscinet/article?mr=0954295 MR0954295] [https://zbmath.org/?q=an%3A0662.14017 Zbl 0662.14017] </td></tr><tr><td valign="top">[a3b]</td> <td valign="top"> V. Kolyvagin, "On the Mordell–Weil group and the Shafarevich–Tate group of Weil elliptic curves" ''Math. USSR Izv.'' , '''33''' (1989) ''Izv. Akad. Nauk SSSR'' , '''52''' (1988) pp. 1154–1180 [https://mathscinet.ams.org/mathscinet/article?mr=984214 MR984214] [https://zbmath.org/?q=an%3A0749.14012 Zbl 0749.14012] </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> V.A. Kolyvagin, "On the structure of the Shafarevich–Tate groups" S. Block (ed.) et al. (ed.) , ''Algebraic geometry'' , ''Lect. notes in math.'' , '''1479''' , Springer (1991) pp. 94–121 [https://mathscinet.ams.org/mathscinet/article?mr=1181210 MR1181210] [https://zbmath.org/?q=an%3A0753.14025 Zbl 0753.14025] </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> K. Rubin, "The Tate–Shafarevich group and $z099/z099260/z099260436.png"$-functions of elliptic curves with complex multiplication" ''Invent. Math.'' , '''89''' (1987) pp. 527–560 </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> S. Bloch, "Algebraic cycles and values of $z099/z099260/z099260437.png"$-functions I" ''J. Reine Angew. Math.'' , '''350''' (1984) pp. 94–108 [https://mathscinet.ams.org/mathscinet/article?mr=743535 MR743535] </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> S. Bloch, "Algebraic cycles and values of $z099/z099260/z099260438.png"$-functions II" ''Duke Math. J.'' , '''52''' (1985) pp. 379–397 [https://mathscinet.ams.org/mathscinet/article?mr=792179 MR792179] </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> A. Beilinson, "Higher regulators and values of $z099/z099260/z099260439.png"$-functions" J. Soviet Math. , 30 (10985) pp. 2036–2070 Itogi Nauk. i Tekhn. Sovr. Probl. Mat. , 24 (1984) pp. 181–238