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Doi–Naganuma Lifting
In mathematics, the Doi–Naganuma lifting is a map from elliptic modular forms to Hilbert modular forms of a real quadratic field, introduced by and . It was a precursor of the base change lifting. It is named for Japanese mathematicians Kōji Doi (土井公二） and Hidehisa Naganuma (長沼英久). See also *Saito–Kurokawa lift In mathematics, the Saito–Kurokawa lift (or lifting) takes elliptic modular forms to Siegel modular forms of degree 2. The existence of this lifting was conjectured in 1977 independently by Hiroshi Saito and . Its existence was almost proved by , ..., a similar lift to Siegel modular forms References * * * {{DEFAULTSORT:Doi-Naganuma lifting Modular forms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Modular Form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a 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 with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic forms which are functions defined on Lie groups which transform nicely with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Modular Form
In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the ''m''-fold product of upper half-planes \mathcal satisfying a certain kind of functional equation. Definition Let ''F'' be a totally real number field of degree ''m'' over the rational field. Let \sigma_1, \ldots, \sigma_m be the real embeddings of ''F''. Through them we have a map :GL_2(F) \to GL_2(\R)^m. Let \mathcal O_F be the ring of integers of ''F''. The group GL_2^+(\mathcal O_F) is called the ''full Hilbert modular group''. For every element z = (z_1, \ldots, z_m) \in \mathcal^m, there is a group action of GL_2^+ (\mathcal O_F) defined by \gamma \cdot z = (\sigma_1(\gamma) z_1, \ldots, \sigma_m(\gamma) z_m) For :g = \begina & b \\ c & d \end \in GL_2(\R), define: :j(g, z) = \det(g)^ (cz+d) A Hilbert modular form of weight (k_1,\ldots,k_m) is an analytic function on \mathcal^m such that for every \gamma \in GL_2^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the s. Quadratic fields have been studied in great depth, initially as part of the theory of s. There remain some unsolved prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base Change Lifting
In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galois group to a subgroup. The Doi–Naganuma lifting from 1967 was a precursor of the base change lifting. Base change lifting was introduced by for Hilbert modular form In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the ''m''-fold product of upper half-planes \mathcal satisfying a certain kind of functional e ...s of cyclic totally real fields of prime degree, by comparing the trace of twisted Hecke operators on Hilbert modular forms with the trace of Hecke operators on ordinary modular forms. gave a representation theoretic interpretation of Saito's results and used this to generalize them. extended the base change lifting to more general automorphic forms and showed how to u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saito–Kurokawa Lift
In mathematics, the Saito–Kurokawa lift (or lifting) takes elliptic modular forms to Siegel modular forms of degree 2. The existence of this lifting was conjectured in 1977 independently by Hiroshi Saito and . Its existence was almost proved by , and and completed the proof. Statement The Saito–Kurokawa lift ''σ''''k'' takes level 1 modular forms ''f'' of weight 2''k'' − 2 to level 1 Siegel modular forms of degree 2 and weight ''k''. The L-functions (when ''f'' is a Hecke eigenforms) are related by ''L''(''s'',''σ''''k''(''f'')) = ζ(''s'' − ''k'' + 2)ζ(''s'' − ''k'' + 1)''L''(''s'', ''f''). The Saito–Kurokawa lift can be constructed as the composition of the following three mappings: # The Shimura correspondence from level 1 modular forms of weight 2''k'' − 2 to a space of level 4 modular forms of weight ''k'' − 1/2 in the Kohnen plus-space. #A map from the Kohnen plus-space t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
