Doubly isogenous genus-2 curves with 𝐷₄-action

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C prime"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are curves over a finite field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational base points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P prime"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">P’</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="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D prime"> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">D’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-<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> maps on their Jacobians. We say that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper C comma upper P right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(C,P)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper C prime comma upper P prime right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(C’,P’)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <italic>doubly isogenous</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J a c left-parenthesis upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>a</mml:mi> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Jac(C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J a c left-parenthesis upper C prime right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>a</mml:mi> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Jac(C’)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isogenous over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J a c left-parenthesis upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>a</mml:mi> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Jac(D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J a c left-parenthesis upper D prime right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>a</mml:mi> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Jac(D’)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isogenous over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For curves of genus <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> whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon.