Abstract
<p> A lot of work has gone into computing images of Galois representations coming from elliptic curves. This article presents an algorithm to determine the image of the mod- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Galois representation associated to a principally polarized abelian surface over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Conjugacy class distribution of subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper S p left-parenthesis 4 comma double-struck upper F 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GSp(4,\mathbb {F}_3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a key ingredient. While this ingredient is feasible to compute for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper S p left-parenthesis 4 comma double-struck upper F Subscript script l Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi> ℓ </mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GSp(4,\mathbb {F}_\ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any small prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , distinguishing Gassmann-equivalent subgroups is a delicate problem. We accomplish it for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l equals 3"> <mml:semantics> <mml:mrow> <mml:mi> ℓ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell = 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using several techniques. The algorithm does not require the knowledge of endomorphisms. </p>