Algorithms for Chow-Heegner points via iterated integrals
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Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript slash bold upper Q"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">E_{/\mathbf Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an elliptic curve of conductor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> newform on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 0 left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma _0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated to it by modularity. Building on an idea of S. Zhang, an article by Darmon, Rotger, and Sols describes the construction of so-called <italic>Chow-Heegner points</italic> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript upper T comma f Baseline element-of upper E left-parenthesis bold upper Q overbar right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ¯ </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P_{T,f}\in E({\bar {\mathbf Q}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , indexed by algebraic correspondences <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T subset-of upper X 0 left-parenthesis upper N right-parenthesis times upper X 0 left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T\subset X_0(N)\times X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It also gives an analytic formula, depending only on the image of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in cohomology under the complex cycle class map, for calculating <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript upper T comma f"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">P_{T,f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> numerically via Chen’s theory of iterated integrals. The present work describes an algorithm based on this formula for computing the Chow-Heegner points to arbitrarily high complex accuracy, carries out the computation for all elliptic curves of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and conductor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than 100"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml
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