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# Ihara's lemma

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 *J*_{0}(*N*)×*J*_{0}(*N*) to *J*_{0}(*Np*) is Eisenstein whenever the prime *p* does not divide *N*. Here *J*_{0}(*N*) is the Jacobian of the compactification of the modular curve of Γ_{0}(*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, MR 0399105

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, MR 804706

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