Local zeta-function
Encyclopedia
In number theory
, a local zeta-function
is a function whose logarithmic derivative
is a generating function
for the number of solutions of a set of equations defined over a finite field
F, in extension fields Fk of F.
, just one field Fk with
,
for k = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety
V — defined over F, we can count the number
of solutions in Fk and create the generating function
.
The correct definition for Z(t) is to make log Z equal to G, and so
we will have Z(0) = 1 since G(0) = 0, and Z(t) is a priori a formal power series
.
Note that the logarithmic derivative
equals the generating function
.
is the expansion of a logarithm (for |t| < 1). In this case we have
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 as we must the one point at infinity. Therefore we shall have
and
for |t| small enough.
In this case we have
The first study of these functions was in the 1923 dissertation of Emil Artin
. He obtained results for the case of 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
. The earliest known non-trivial cases of local zeta-functions were implicit in Carl Friedrich Gauss
's Disquisitiones Arithmeticae
, article 385; there certain particular examples of elliptic curve
s over finite fields having complex multiplication
have their points counted by means of cyclotomy.
of t, something that is interesting even in the case of V an elliptic curve
over finite field.
It is the functions Z that are designed to multiply, to get global zeta functions. Those involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime number
s). In that connection, the variable t undergoes substitution by p-s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse-Weil zeta-function
.)
With that understanding, the products of the Z in the two cases used as examples come out as
and 
.
with P(t) a polynomial, of degree 2g where g is the genus
of C. The Riemann hypothesis for curves over finite fields states that the roots of P have absolute value
where q = |F|.
For example, for the elliptic curve case there are two roots, and it is easy to show their product is q−1. 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 writing up the algebraic geometry
involved. This led him to the general Weil conjectures
, finally proved a generation later. See étale cohomology
for the basic formulae of the general theory.
Here
is a separated scheme of finite type over the finite field F with 
elements, and Frobq is the geometric Frobenius acting on 
-adic étale cohomology with compact supports of 
, the lift of 
to the algebraic closure of the field F. This shows that the zeta function is a rational function of 
.
An infinite product formula for
is
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
, 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
.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a local zeta-function
- Z(t)
is a function whose logarithmic derivative
Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
is a generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
for the number of solutions of a set of equations defined over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
F, in extension fields Fk of F.
Formulation
Given F, there is, up to isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
, just one field Fk with
,for k = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V — defined over F, we can count the number
of solutions in Fk and create the generating function
.The correct definition for Z(t) is to make log Z equal to G, and so
we will have Z(0) = 1 since G(0) = 0, and Z(t) is a priori a formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
.
Note that the logarithmic derivative
Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
equals the generating function
.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 a point. Then- G(t) = −log(1 − t)
is the expansion of a logarithm (for |t| < 1). In this case we have
- Z(t) = 1/(1 − t).
To take something more interesting, let V be the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
over F. If F has q elements, then this has q + 1 points, including as we must the one point at infinity. Therefore we shall have
- Nk = qk + 1
and
- G(t) = −log(1 − t) − log(1 − qt),
for |t| small enough.
In this case we have
- Z(t) = 1/{(1 − t)(1 − qt)}.
The first study of these functions was in the 1923 dissertation of Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
. He obtained results for the case of 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
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
. The earliest known non-trivial cases of local zeta-functions were implicit in Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
's Disquisitiones Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
, article 385; there certain particular examples of elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s over finite fields having complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
have their points counted by means of cyclotomy.
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 functionRational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
of t, something that is interesting even in the case of V an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
over finite field.
It is the functions Z that are designed to multiply, to get global zeta functions. Those involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s). In that connection, the variable t undergoes substitution by p-s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse-Weil zeta-function
Hasse-Weil zeta function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function. Such L-functions are called 'global', in that they are defined as Euler products in terms of local zeta functions...
.)
With that understanding, the products of the Z in the two cases used as examples come out as
and .Riemann hypothesis for curves over finite fields
For projective curves C over F that are non-singular, it can be shown that- Z(t) = P(t)/{(1 − t)(1 − qt)},
with P(t) a polynomial, of degree 2g where g is the genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
of C. The Riemann hypothesis for curves over finite fields states that the roots of P have absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
- q−1/2,
where q = |F|.
For example, for the elliptic curve case there are two roots, and it is easy to show their product is q−1. Hasse's theorem is that they have the same absolute value; and this has immediate consequences for the number of points.
André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
proved this for the general case, around 1940 (Comptes Rendus note, April 1940): he spent much time in the years after that writing up the algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
involved. This led him to the general Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
, finally proved a generation later. See étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
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 thatHere
is a separated scheme of finite type over the finite field F with elements, and Frobq is the geometric Frobenius acting on -adic étale cohomology with compact supports of , the lift of to the algebraic closure of the field F. This shows that the zeta function is a rational function of .An infinite product formula for
isHere, 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
, where two points are equivalent if they are conjugates over F. The degree of x is the degree of the field extension of Fgenerated 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
.