Ihara's lemma

Short description: On when the kernel of the sum of the two p-degeneracy maps is Eisenstein

In mathematics, Ihara's lemma, introduced by Ihara (1975, lemma 3.2) and named by (Ribet 1984), states that the kernel of the sum of the two p-degeneracy maps from J₀(N)×J₀(N) to J₀(Np) is Eisenstein whenever the prime p does not divide N. Here J₀(N) is the Jacobian of the compactification of the modular curve of Γ₀(N).

References

• Ihara, Yasutaka (1975), "On modular curves over finite fields", in Baily, Walter L., Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), Tata Institute of Fundamental Research Studies in Mathematics, 7, Oxford University Press, pp. 161–202, ISBN 978-0-19-560525-9 
• Ribet, Kenneth A. (1984), "Congruence relations between modular forms", Proceedings of the International Congress of Mathematicians, Vol. 1 (Warsaw, 1983), Warszawa: PWN, pp. 503–514, http://ada00.math.uni-bielefeld.de/ICM/ICM1983.1/, retrieved 2012-11-09