2A01:E0A:2D4:37B0:9963:6D62:64E2:1439 at 00:25, 10 February 2025

2025-02-10T00:25:30Z

New page

In [[mathematics]], the '''local zeta function''' {{math|''Z''(''V'', ''s'')}} (sometimes called the '''congruent zeta function''' or the [[Hasse–Weil zeta function]]) is defined as

:<math>Z(V, s) = \exp\left(\sum_{k = 1}^\infty \frac{N_k}{k} (q^{-s})^k\right)</math>

where {{mvar|V}} is a [[Singular point of an algebraic variety|non-singular]] {{mvar|n}}-dimensional [[projective algebraic variety]] over the field {{math|'''F'''''q''}} with {{mvar|q}} elements and {{math|''N''''k''}} is the number of points of {{mvar|''V''}} defined over the finite field extension {{math|'''F'''''q''''k''}} of {{math|'''F'''''q''}}.<ref>Section V.2 of {{Citation
| last=Silverman
| first=Joseph H.
| author-link=Joseph H. Silverman
| title=The arithmetic of elliptic curves
| publisher=[[Springer-Verlag]]
| location=New York
| series=[[Graduate Texts in Mathematics]]
| isbn=978-0-387-96203-0
| mr=1329092
| year=1992
| volume=106
}}</ref>

Making the variable transformation {{math|''t'' {{=}} ''q''−''s'',}} gives
:<math>
\mathit{Z} (V,t) = \exp
\left( \sum_{k=1}^{\infty} N_k \frac{t^k}{k} \right)
</math>
as the [[formal power series]] in the variable <math>t</math>.

Equivalently, the local zeta function is sometimes defined as follows:
:<math>
(1)\ \ \mathit{Z} (V,0) = 1 \,
</math>
:<math>
(2)\ \ \frac{d}{dt} \log \mathit{Z} (V,t) = \sum_{k=1}^{\infty} N_k t^{k-1}\ .</math>

In other words, the local zeta function {{math|''Z''(''V'', ''t'')}} with coefficients in the [[finite field]] {{math|'''F'''''q''}} is defined as a function whose [[logarithmic derivative]] generates the number {{math|''N''''k''}} of solutions of the equation defining {{mvar|V}} in the degree {{mvar|k}} extension {{math|'''F'''''q''''k''.}}



==Formulation==

Given a finite field ''F'', there is, up to [[isomorphism]], only one field ''Fk'' with

:<math>[ F_k : F ] = k \,</math>,

for ''k'' = 1, 2, ... . When ''F'' is the unique field with ''q'' elements, ''Fk'' is the unique field with <math>q^k</math> elements. Given a set of polynomial equations — or an [[algebraic variety]] ''V'' — defined over ''F'', we can count the number

:<math>N_k \,</math>

of solutions in ''Fk'' and create the generating function

:<math>G(t) = N_1t +N_2t^2/2 + N_3t^3/3 +\cdots \,</math>.

The correct definition for ''Z''(''t'') is to set log ''Z'' equal to ''G'', so

:<math>Z= \exp (G(t)) \, </math>

and ''Z''(0) = 1, since ''G''(0) = 0, and ''Z''(''t'') is ''a priori'' a [[formal power series]].

The [[logarithmic derivative]]

:<math>Z'(t)/Z(t) \,</math>

equals the generating function

:<math>G'(t) = N_1 +N_2t^1 + N_3t^2 +\cdots \,</math>.

==Examples==

For example, assume all the ''Nk'' are 1; this happens for example if we start with an equation like ''X'' = 0, so that geometrically we are taking ''V'' to be a point. Then

:<math>G(t) = -\log(1 - t)</math>

is the expansion of a logarithm (for |''t''| < 1). In this case we have

:<math>Z(t) = \frac{1}{(1 - t)}\ .</math>

To take something more interesting, let ''V'' be the [[projective line]] over ''F''. If ''F'' has ''q'' elements, then this has ''q'' + 1 points, including the one [[point at infinity]]. Therefore, we have

:<math>N_k = q^k + 1</math>

and

:<math>G(t) = -\log(1 - t) -\log(1 - qt)</math>

for |''t''| small enough, and therefore

:<math>Z(t) = \frac{1}{(1 - t)(1 - qt)}\ .</math>

The first study of these functions was in the 1923 dissertation of [[Emil Artin]]. He obtained results for the case of a [[hyperelliptic curve]], and conjectured the further main points of the theory as applied to curves. The theory was then developed by [[F. K. Schmidt]] and [[Helmut Hasse]].<ref>[[Daniel Bump]], ''Algebraic Geometry'' (1998), p. 195.</ref> The earliest known nontrivial cases of local zeta functions were implicit in [[Carl Friedrich Gauss]]'s ''[[Disquisitiones Arithmeticae]]'', article 358. There, certain particular examples of [[elliptic curve]]s over finite fields having [[complex multiplication]] have their points counted by means of [[cyclotomy]].<ref>[[Barry Mazur]], ''Eigenvalues of Frobenius'', p. 244 in ''Algebraic Geometry, Arcata 1974: Proceedings American Mathematical Society'' (1974).</ref>

For the definition and some examples, see also.<ref>[[Robin Hartshorne]], ''Algebraic Geometry'', p. 449 Springer 1977 APPENDIX C "The Weil Conjectures"</ref>

==Motivations==

The relationship between the definitions of ''G'' and ''Z'' can be explained in a number of ways. (See for example the infinite product formula for ''Z'' below.) In practice it makes ''Z'' a [[rational function]] of ''t'', something that is interesting even in the case of ''V'' an [[elliptic curve]] over a finite field.

The local ''Z'' zeta functions are multiplied to get global ''<math>\zeta</math>'' zeta functions,

<math>\zeta = \prod Z</math>

These generally involve different finite fields (for example the whole family of fields '''Z'''/''p'''''Z''' as ''p'' runs over all [[prime number]]s).

In these fields, the variable ''t'' is substituted by ''p−s'', where ''s'' is the complex variable traditionally used in [[Dirichlet series]]. (For details see [[Hasse–Weil zeta function]].)

The global products of ''Z'' in the two cases used as examples in the previous section therefore come out as <math>\zeta(s)</math> and <math>\zeta(s)\zeta(s-1)</math> after letting <math>q=p</math>.

==Riemann hypothesis for curves over finite fields==

For projective curves ''C'' over ''F'' that are [[Algebraic curve#Singularities|non-singular]], it can be shown that

:<math>Z(t) = \frac{P(t)}{(1 - t)(1 - qt)}\ ,</math>

with ''P''(''t'') a polynomial, of degree 2''g'', where ''g'' is the [[genus (mathematics)|genus]] of ''C''. Rewriting

:<math>P(t)=\prod^{2g}_{i=1}(1-\omega_i t)\ ,</math>

the '''Riemann hypothesis for curves over finite fields''' states

:<math>|\omega_i|=q^{1/2}\ .</math>

For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are ''q''1/2. [[Hasse's theorem on elliptic curves|Hasse's theorem]] is that they have the same absolute value; and this has immediate consequences for the number of points.

[[André Weil]] proved this for the general case, around 1940 (''Comptes Rendus'' note, April 1940): he spent much time in the years after that [[Foundations of Algebraic Geometry|writing]] up the [[algebraic geometry]] involved. This led him to the general [[Weil conjectures]]. [[Alexander Grothendieck]] developed [[scheme (mathematics)|scheme]] theory for the purpose of resolving these.
A generation later [[Pierre Deligne]] completed the proof.
(See [[étale cohomology]] for the basic formulae of the general theory.)

==General formulas for the zeta function==

It is a consequence of the [[Lefschetz trace formula]] for the [[Frobenius morphism]] that

:<math>Z(X,t)=\prod_{i=0}^{2\dim X}\det\big(1-t \mbox{Frob}_q |H^i_c(\overline{X},{\mathbb Q}_\ell)\big)^{(-1)^{i+1}}.</math>

Here <math>X</math> is a separated scheme of finite type over the finite field ''F'' with <math>q</math> elements, and Frobq is the geometric Frobenius acting on <math>\ell</math>-adic étale cohomology with compact supports of <math>\overline{X}</math>, the lift of <math>X</math> to the algebraic closure of the field ''F''. This shows that the zeta function is a rational function of <math>t</math>.

An infinite product formula for <math>Z(X, t)</math> is

:<math>Z(X, t)=\prod\ (1-t^{\deg(x)})^{-1}.</math>

Here, the product ranges over all closed points ''x'' of ''X'' and deg(''x'') is the degree of ''x''.
The local zeta function ''Z(X, t)'' is viewed as a function of the complex variable ''s'' via the change of
variables ''q−s''.

In the case where ''X'' is the variety ''V'' discussed above, the closed points
are the equivalence classes ''x=[P]'' of points ''P'' on <math>\overline{V}</math>, where two points are equivalent if they are conjugates over ''F''. The degree of ''x'' is the degree of the field extension of ''F''
generated by the coordinates of ''P''. The logarithmic derivative of the infinite product ''Z(X, t)'' is easily seen to be the generating function discussed above, namely

:<math>N_1 +N_2t^1 + N_3t^2 +\cdots \,</math>.

==See also==
*[[List of zeta functions]]
*[[Weil conjectures]]
*[[Elliptic curve]]

==References==
{{reflist}}

{{Bernhard Riemann}}

[[Category:Algebraic varieties]]
[[Category:Finite fields]]
[[Category:Diophantine geometry]]
[[Category:Zeta and L-functions]]
[[Category:Fixed points (mathematics)]]
[[Category:Bernhard Riemann]]

Local zeta function - Revision history

2A01:E0A:2D4:37B0:9963:6D62:64E2:1439 at 00:25, 10 February 2025