Hecke Algebra Of A Locally Compact Group
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Erich Hecke (; 20 September 1887 – 13 February 1947) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
known for his work in
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and the theory of
modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
s.


Biography

Hecke was born in
Buk Buk or BUK may refer to: Places Czech Republic * Buk (Prachatice District), a municipality and village in the South Bohemian Region * Buk (Přerov District), a municipality and village in the Olomouc Region *Buk, a village and part of Jindřichů ...
,
Province of Posen The Province of Posen (; ) was a Provinces of Prussia, province of the Kingdom of Prussia from 1848 to 1920, occupying most of the historical Greater Poland. The province was established following the Greater Poland Uprising (1848), Poznań Uprisi ...
,
German Empire The German Empire (),; ; World Book, Inc. ''The World Book dictionary, Volume 1''. World Book, Inc., 2003. p. 572. States that Deutsches Reich translates as "German Realm" and was a former official name of Germany. also referred to as Imperia ...
(now
Poznań Poznań ( ) is a city on the Warta, River Warta in west Poland, within the Greater Poland region. The city is an important cultural and business center and one of Poland's most populous regions with many regional customs such as Saint John's ...
,
Poland Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
). He obtained his doctorate in
Göttingen Göttingen (, ; ; ) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. According to the 2022 German census, t ...
under the supervision of
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
.
Kurt Reidemeister Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttin ...
and
Heinrich Behnke Heinrich Adolph Louis Behnke (9 October 1898 in Horn – 10 October 1979 in Münster) was a German mathematician and rector at the University of Münster. Life and career He was born into a Lutheran family in Horn, a suburb of Hamburg. He att ...
were among his students. In 1933 Hecke signed the '' Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State'', but was later known as being opposed to the Nazis. Hecke died in
Copenhagen Copenhagen ( ) is the capital and most populous city of Denmark, with a population of 1.4 million in the Urban area of Copenhagen, urban area. The city is situated on the islands of Zealand and Amager, separated from Malmö, Sweden, by the ...
,
Denmark Denmark is a Nordic countries, Nordic country in Northern Europe. It is the metropole and most populous constituent of the Kingdom of Denmark,, . also known as the Danish Realm, a constitutionally unitary state that includes the Autonomous a ...
.
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...
, in the foreword to his text ''
Basic Number Theory ''Basic Number Theory'' is an influential book by André Weil, an exposition of algebraic number theory and class field theory with particular emphasis on valuation-theoretic methods. Based in part on a course taught at Princeton University in 1 ...
'' says: "To improve upon Hecke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task", referring to Hecke's book "Lectures on the Theory of Algebraic Numbers."


Research

His early work included establishing the
functional equation 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 ...
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 ...
, with a proof based on
theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube ...
s. The method extended to the
L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...
s associated to a class of characters now known as
Hecke 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 ...
s or
idele class character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-function, ''L''-functions larger than Dirichlet L-function, Dirichlet ''L''-functions, and a natural setting for t ...
s; such L-functions are now known as Hecke L-functions. He devoted most of his research to the theory of
modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
s, creating the general theory of
cusp form In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. Introduction A cusp form is distinguished in the case of modular forms for the modular gr ...
s (
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
, for GL(2)), as it is now understood in the classical setting.


Recognition

He was a Plenary Speaker of the
ICM ICM may refer to: Organizations * Irish Church Missions, an Anglican mission * Institut du Cerveau, the Paris Brain Institute, a research center * Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw * Interna ...
in 1936 in Oslo.


See also

*
List of things named after Erich Hecke {{Short description, none These are things named after Erich Hecke, a German mathematician. * Hecke algebra ** Hecke algebra of a locally compact group **Hecke algebra of a finite group **Hecke algebra of a pair ** Hecke polynomial **Iwahori–Heck ...
*
Hecke algebra (disambiguation) In mathematics, a Hecke algebra is classically the algebra of Hecke operators studied by Erich Hecke. It may also refer to one of several algebras generalizing the classical Hecke algebra: * Hecke algebra of a pair (''G'',''K'') where ''G'' is a r ...
*
Tate's thesis In number theory, Tate's thesis is the 1950 PhD thesis of completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta functi ...


References


External links

* * {{DEFAULTSORT:Hecke, Erich 1887 births 1947 deaths People from Poznań County People from the Province of Posen 20th-century German mathematicians German number theorists Academic staff of the University of Hamburg Academic staff of the University of Basel Academic staff of the University of Göttingen University of Breslau alumni University of Göttingen alumni Presidents of the German Mathematical Society