Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves

Abstract

We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j left-parenthesis upper E right-parenthesis not-equals 0 comma 1728"> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≠ </mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1728</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">j(E)\neq 0, 1728</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="q identical-to 1 left-parenthesis mod 12 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo> ≡ </mml:mo> <mml:mn>1</mml:mn> <mml:mspace width="0.667em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em"/> <mml:mn>12</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">q\equiv 1 \pmod {12}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of special values of Gaussian hypergeometric series. This paper uses methods introduced by Fuselier for one-parameter families of curves to express the trace of Frobenius of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a function of its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant and discriminant instead of a parameter, which are more intrinsic characteristics of the curve.