In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a Hecke character is a generalisation of a
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \chi ...
, introduced by
Erich Hecke
Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms.
Biography
Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He ...
to construct a class of
''L''-functions larger than
Dirichlet ''L''-functions, and a natural setting for the
Dedekind zeta-function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
s and certain others which have
functional equations
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
analogous to that of the
Riemann zeta-function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
A name sometimes used for ''Hecke character'' is the German term Größencharakter (often written Grössencharakter, Grossencharacter, etc.).
Definition using ideles
A Hecke character is a
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of the
idele class group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...
of a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
or
global function field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global function fi ...
. It corresponds uniquely to a character of the
idele group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; t ...
which is trivial on
principal ideles, via composition with the projection map.
This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a "quasicharacter"), or as a homomorphism to the
unit circle in C ("unitary"). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions.
The conductor of a Hecke character ''χ'' is the largest ideal ''m'' such that ''χ'' is a Hecke character mod ''m''. Here we say that ''χ'' is a Hecke character mod ''m'' if ''χ'' (considered as a character on the idele group) is trivial on the group of finite ideles whose every v-adic component lies in 1 + ''m''O
v.
Definition using ideals
The original definition of a Hecke character, going back to Hecke, was in terms of
a character on
fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral doma ...
s. For a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
''K'', let
''m'' = ''m''
''f''''m''
∞ be a
''K''-
modulus, with ''m''
''f'', the "finite part", being an integral ideal of ''K'' and ''m''
∞, the "infinite part", being a (formal) product of real
place
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own Municipality, municipal government
* "Place", a type of street or road ...
s of ''K''. Let ''I''
''m''
denote the group of fractional ideals of ''K'' relatively prime to ''m''
''f'' and
let ''P''
''m'' denote the subgroup of principal fractional ideals (''a'')
where ''a'' is near 1 at each place of ''m'' in accordance with the multiplicities of
its factors: for each finite place ''v'' in ''m''
''f'', ord
''v''(''a'' − 1) is at least as large as the exponent for ''v'' in ''m''
''f'', and ''a'' is positive under each real embedding in ''m''
∞. A Hecke character with modulus ''m''
is a group homomorphism from ''I''
''m'' into the nonzero complex numbers
such that on ideals (''a'') in ''P''
''m'' its value is equal to the
value at ''a'' of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of ''K'' where each local component of the homomorphism has the same real part (in the exponent). (Here we embed ''a'' into the product of Archimedean completions of ''K'' using embeddings corresponding to the various Archimedean places on ''K''.) Thus a Hecke character may be defined on the
ray class group In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields.
The te ...
modulo ''m'', which is the quotient ''I''
''m''/''P''
''m''.
Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked with moduli where all real places appeared.
The role of the infinite part ''m''
∞ is now subsumed under the notion of
an infinity-type.
Relationship between the definitions
The ideal definition is much more complicated than the idelic one, and Hecke's motivation for his definition was to construct ''L''-functions (sometimes referred to as Hecke ''L''-functions) that extend the notion of a Dirichlet ''L''-function from the rationals to other number fields. For a Hecke character χ, its ''L''-function is defined to be the
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
:
carried out over integral ideals relatively prime to the modulus ''m'' of the Hecke character.
The notation ''N(I)'' means the
ideal norm In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal ...
. The common real part condition governing the behavior of Hecke characters on the subgroups ''P''
''m'' implies these
Dirichlet series are absolutely convergent in some right half-plane. Hecke proved these ''L''-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at ''s'' = 1 when the character is trivial. For primitive Hecke characters (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these ''L''-functions satisfy a functional equation relating the values of the ''L''-function of a character and the ''L''-function of its complex conjugate character.
Consider a character ψ of the idele class group, taken to be a map into the unit circle which is 1 on principal ideles and on an exceptional finite set ''S'' containing all infinite places. Then ψ generates a character χ of the ideal group ''I''
''S'', the free abelian group on the prime ideals not in ''S''.
[Heilbronn (1967) p.204] Take a uniformising element π for each prime p not in ''S'' and define a map Πfrom ''I''
''S'' to idele classes by mapping each p to the class of the idele which is π in the p coordinate and 1 everywhere else. Let χ be the composite of Πand ψ. Then χ is well-defined as a character on the ideal group.
[Heilbronn (1967) p. 205]
In the opposite direction, given an ''admissible'' character χ of ''I''
''S'' there corresponds a unique idele class character ψ. Here admissible refers to the existence of a modulus m based on the set ''S'' such that the character χ is 1 on the ideals which are 1 mod m.
[Heilbronn (1967) p.207]
The characters are 'big' in the sense that the infinity-type when present non-trivially means these characters are not of finite order. The finite-order Hecke characters are all, in a sense, accounted for by
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
: their ''L''-functions are
Artin ''L''-functions, as
Artin reciprocity
Artin may refer to:
* Artin (name), a surname and given name, including a list of people with the name
** Artin, a variant of Harutyun
Harutyun ( hy, Õ€Õ¡Ö€Õ¸Ö‚Õ©ÕµÕ¸Ö‚Õ¶ and in Western Armenian Õ…Õ¡Ö€Õ¸Ö‚Õ©Õ«Ö‚Õ¶) also spelled Haroutioun, Harut ...
shows. But even a field as simple as the
Gaussian field has Hecke characters that go beyond finite order in a serious way (see the example below). Later developments in
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
theory indicated that the proper place of the 'big' characters was to provide the
Hasse–Weil ''L''-functions for an important class of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
(or even
motive
Motive(s) or The Motive(s) may refer to:
* Motive (law)
Film and television
* ''Motives'' (film), a 2004 thriller
* ''The Motive'' (film), 2017
* ''Motive'' (TV series), a 2013 Canadian TV series
* ''The Motive'' (TV series), a 2020 Israeli T ...
s).
Special cases
*A
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \chi ...
is a Hecke character of finite order. It is determined by values on the set of totally positive principal ideals which are 1 with respect to some modulus m.
[
*A ]Hilbert character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ...
is a Dirichlet character of conductor 1.[ The number of Hilbert characters is the order of the class group of the field. Class field theory identifies the Hilbert characters with the characters of the Galois group of the Hilbert class field.
]
Examples
*For the field of rational numbers, the idele class group is isomorphic to the product of positive reals
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
â„+ with all the unit groups of the ''p''-adic integers. So a quasicharacter can be written as product of a power of the norm with a Dirichlet character.
*A Hecke character χ of the Gaussian integers of conductor 1 is of the form
: χ((''a'')) = , ''a'', ''s''(''a''/, ''a'', )4''n''
:for ''s'' imaginary and ''n'' an integer, where ''a'' is a generator of the ideal (''a''). The only units are powers of ''i'', so the factor of 4 in the exponent ensures that the character is well defined on ideals.
Tate's thesis
Hecke's original proof of the functional equation for ''L''(''s'',χ) used an explicit theta-function. John Tate John Tate may refer to:
* John Tate (mathematician) (1925–2019), American mathematician
* John Torrence Tate Sr. (1889–1950), American physicist
* John Tate (Australian politician) (1895–1977)
* John Tate (actor) (1915–1979), Australian act ...
's 1950 Princeton doctoral dissertation, written under the supervision of Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
, applied Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
systematically, to remove the need for any special functions. A similar theory was independently developed by Kenkichi Iwasawa
Kenkichi Iwasawa ( ''Iwasawa Kenkichi'', September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory.
Biography
Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gun ...
which was the subject of his 1950 ICM talk. A later reformulation in a Bourbaki seminar by showed that parts of Tate's proof could be expressed by distribution theory: the space of distributions (for Schwartz–Bruhat test functions) on the adele group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...
of ''K'' transforming under the action of the ideles by a given χ has dimension 1.
Algebraic Hecke characters
An algebraic Hecke character is a Hecke character taking algebraic values: they were introduced by Weil in 1947 under the name type A0. Such characters occur in class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
and the theory of complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
.
Indeed let ''E'' be an elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
defined over a number field ''F'' with complex multiplication by the imaginary quadratic field ''K'', and suppose that ''K'' is contained in ''F''. Then there is an algebraic Hecke character χ for ''F'', with exceptional set ''S'' the set of primes of bad reduction
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of ...
of ''E'' together with the infinite places. This character has the property that for a prime ideal p of good reduction, the value χ(p) is a root of the characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
of the Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
. As a consequence, the 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 a meromorphic function on the complex plane defined in terms of the number of points on the variety after reduc ...
for ''E'' is a product of two Dirichlet series, for χ and its complex conjugate.[Husemoller (1987) pp. 302–303; (2002) pp. 321–322]
Notes
References
*
*
*
*
*
*
*J. Tate, ''Fourier analysis in number fields and Hecke's zeta functions'' (Tate's 1950 thesis), reprinted in ''Algebraic Number Theory'' edd J. W. S. Cassels, A. Fröhlich
A is the first letter of the Latin and English alphabet.
A may also refer to:
Science and technology Quantities and units
* ''a'', a measure for the attraction between particles in the Van der Waals equation
* ''A'' value, a measure of ...
(1967) pp. 305–347.
*
*
{{DEFAULTSORT:Hecke Character
Number theory
Zeta and L-functions