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The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent (k − 1)/2 where k is the weight of the form.
In mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan: (+) ...
Then, the Ramanujan bounds of W. Luo, Z. Rudnick and P. Sarnak [11] for GL(N) over number fields yield non-trivial bounds for the generalized Ramanujan conjecture of the classical groups. Symmetric powers for GL(2) : Proofs of functoriality for the symmetric cube and for the symmetric fourth [ 12 ] powers of cuspidal automorphic representations ...
Conjecture Field Comments Eponym(s) Cites 1/3–2/3 conjecture: order theory: n/a: 70 abc conjecture: number theory: ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1 ⇒Erdős–Woods conjecture, Fermat–Catalan conjecture Formulated by David Masser and Joseph Oesterlé. [1] Proof claimed in 2012 by Shinichi Mochizuki: n/a ...
Lafforgue's theorem implies the Ramanujan–Petersson conjecture that if an automorphic form for GL n (F) has central character of finite order, then the corresponding Hecke eigenvalues at every unramified place have absolute value 1.
The generalized Ramanujan conjecture for the general linear group implies Selberg's conjecture. More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL 2 over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series ...
Flicker, Yuval Z.; Kazhdan, David A. (1989), Geometric Ramanujan conjecture and Drinfeld reciprocity law, Number theory, trace formulas and discrete groups, Symp. in Honor of Atle Selberg, Oslo/Norway 1987, 201-218 (1989).
Elliott–Halberstam conjecture; Functional equation (L-function) Chebotarev's density theorem; Local zeta function. Weil conjectures; Modular form. modular group; Congruence subgroup; Hecke operator; Cusp form; Eisenstein series; Modular curve; Ramanujan–Petersson conjecture; Birch and Swinnerton-Dyer conjecture; Automorphic form; Selberg ...