Shimura curves of genus at most two

Abstract

We enumerate all Shimura curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 Superscript German upper D Baseline left-parenthesis German upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X^{\mathfrak {D}}_0(\mathfrak {N})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of genus at most two: there are exactly 858 such curves, up to equivalence.