Erich Hecke
Encyclopedia
Erich Hecke was a German
mathematician
. He obtained his doctorate in Göttingen under the supervision of David Hilbert
. Kurt Reidemeister
and Heinrich Behnke
were among his students.
Hecke was born in Buk
, Posen
, Germany
(now Poznań
, Poland
), and died in Copenhagen
, Denmark
. His early work included establishing the functional equation
for the Dedekind zeta function
, with a proof based on theta functions. The method extended to the L-function
s associated to a class of characters now known as Hecke character
s or idele class characters: such L-functions are now known as Hecke L-functions. He devoted most of his research to the theory of modular form
s, creating the general theory of cusp form
s (holomorphic, for GL(2)), as it is now understood in the classical setting.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He obtained his doctorate in Göttingen under the supervision of David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. Kurt Reidemeister
Kurt Reidemeister
Kurt Werner Friedrich Reidemeister was a mathematician born in Braunschweig , Germany.He received his doctorate in 1921 with a thesis in algebraic number theory at the University of Hamburg under the supervision of Erich Hecke. In 1923 he was appointed assistant professor at the University of Vienna...
and Heinrich Behnke
Heinrich Behnke
Heinrich Behnke was a German mathematician and rector at the University of Münster.- Life and career :...
were among his students.
Hecke was born in Buk
Buk
Buk is a town in central Poland, situated in the Greater Poland Voivodeship , previously in Poznań Voivodeship .The town's name means "Beech" in Polish, and the flag of the town shows a branch of beech, and three beech leaves....
, Posen
Poznan
Poznań is a city on the Warta river in west-central Poland, with a population of 556,022 in June 2009. It is among the oldest cities in Poland, and was one of the most important centres in the early Polish state, whose first rulers were buried at Poznań's cathedral. It is sometimes claimed to be...
, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
(now Poznań
Poznan
Poznań is a city on the Warta river in west-central Poland, with a population of 556,022 in June 2009. It is among the oldest cities in Poland, and was one of the most important centres in the early Polish state, whose first rulers were buried at Poznań's cathedral. It is sometimes claimed to be...
, Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
), and died in Copenhagen
Copenhagen
Copenhagen is the capital and largest city of Denmark, with an urban population of 1,199,224 and a metropolitan population of 1,930,260 . With the completion of the transnational Øresund Bridge in 2000, Copenhagen has become the centre of the increasingly integrating Øresund Region...
, Denmark
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...
. His early work included establishing the functional equation
Functional equation (L-function)
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural...
for the Dedekind zeta function
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...
, with a proof based on theta functions. The method extended to the L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s associated to a class of characters now known as Hecke character
Hecke character
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to...
s or idele class characters: such L-functions are now known as Hecke L-functions. He devoted most of his research to the theory of modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s, creating the general theory of cusp form
Cusp form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
s (holomorphic, for GL(2)), as it is now understood in the classical setting.
See also
- Hecke algebraHecke algebraIn mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a one-parameter deformation of the group algebra of a Coxeter group....
- Hecke characterHecke characterIn number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to...
- Hecke group
- Hecke L-function
- Hecke operatorHecke operatorIn mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....
- Tate's thesisTate's thesisIn number theory, Tate's thesis is the thesis of where he reformulated the work of Erich Hecke on L-series in terms of Fourier analysis on adelic groups....