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.
42.
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.
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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.
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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.
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The "'generalized Ramanujan conjecture "'or "'Ramanujan Petersson conjecture "', introduced by, is a generalization to other modular forms or automorphic forms.
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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.
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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 ).
48.
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.
49.
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.
50.
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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