automorphic form वाक्य
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- The quotient of a semisimple Lie group by a compact subgroup is a symmetric space and so the theory of automorphic forms is intimately related to harmonic analysis on symmetric spaces.
- Other automorphic forms associated to these congruence subgroups are the holomorphic modular forms, which can be interpreted as cohomology classes on the associated Riemann surfaces via the Eichler-Shimura isomorphism.
- In the case where S = \ Gamma \ setminus \ mathbb H ^ 2 is arithmetic these eigenfunctions are a special type of automorphic forms for \ Gamma called Maass forms.
- Indeed, the L-functions of automorphic forms satisfy the Euler product ( 1 ) but they do not satisfy ( 2 ) because they do not have the completely multiplicative property.
- The "'generalized Ramanujan conjecture "'or "'Ramanujan Petersson conjecture "', introduced by, is a generalization to other modular forms or automorphic forms.
- He is known for his work on automorphic forms and representation theory in the context of the Langlands program, especially the theory of theta correspondence and the Gan-Gross-Prasad conjectures.
- The functoriality conjecture states that a suitable homomorphism of L-groups is expected to give a correspondence between automorphic forms ( in the global case ) or representations ( in the local case ).
- Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups.
- Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence.
- Important results in the theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann-Roch theorem could be applied to calculate the dimension of the space of automorphic forms.
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