isogeny

2002, Mireille Fouquet, François Morain, Isogeny Volcanoes and the SEA Algorithm, Claus Fieker, David R. Kohel (editors), Algorithmic Number Theory: 5th International Symposium, Proceedings, Springer, LNCS 2369, page 279, Lemma 2.2 Let E be an elliptic curve such that Z [π] is maximal at 𝓁. If there exists an 𝓁-isogeny of E, then this 𝓁-isogeny is an horizontal 𝓁-isogeny.

2005, Fred Diamond, Jerry Shurman, A First Course in Modular Forms, Springer, page 29, The dual isogeny of an isomorphism is its inverse. The dual of a composition of isogenies is the composition of the duals in the reverse order. If φ is an isogeny and ̂φ is its dual then the formulas φ(z+Λ)=mz+Λ', φ(z'+Λ')=( operatorname φ/m)z'+Λ show that also φ∘̂φ=[ operatorname deg(φ)]=[ operatorname deg(̂φ)], so that φ is in turn the dual isogeny of its dual ̂φ. Isogeny of complex tori, rather than isomorphism, will turn out to be the appropriate equivalence relation in the context of modular forms.