Isogeny relations in products of families of elliptic curves

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>E</m:mi> <m:mi>λ</m:mi> </m:msub> </m:math> E_{\lambda} be the Legendre family of elliptic curves with equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>Y</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>−</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> Y^{2}=X(X-1)(X-\lambda) . Given a curve 𝒞, satisfying a condition on the degrees of some of its coordinates and parametrizing 𝑚 points <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>P</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>m</m:mi> </m:msub> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi>E</m:mi> <m:mi>λ</m:mi> </m:msub> </m:mrow> </m:math> P_{1},\ldots,P_{m}\in E_{\lambda} and 𝑛 points <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>Q</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>Q</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi>E</m:mi> <m:mi>μ</m:mi> </m:msub> </m:mrow> </m:math> Q_{1},\ldots,Q_{n}\in E_{\mu} and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many points <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">c</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> \mathbf{c}_{0} on 𝒞 such that there exists an isogeny <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ϕ</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:msub> <m:mi>E</m:mi> <m:mrow> <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="bold">c</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo stretchy="false">→</m:mo> <m:msub> <m:mi>E</m:mi> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="bold">c</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mrow> </m:mrow> </m:math> \phi\colon E_{\mu(\mathbf{c}_{0})}\rightarrow E_{\lambda(\mathbf{c}_{0})} and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>+</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> m+n points <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>P</m:mi> <m:mn>