On modular categories with Frobenius–Perron dimension congruent to 2 modulo 4

We contribute to the classification of modular categories <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</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 F upper P d i m left-parenthesis script upper C right-parenthesis identical-to 2 left-parenthesis mod 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>P</mml:mi> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≡ </mml:mo> <mml:mn>2</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>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">FPdim(\mathcal {C})\equiv 2 \pmod 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that such categories have group of invertibles of even order and that they factorize as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C approximately-equals script upper C overTilde squared-times script upper P"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo> ≅ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mo> ⊠ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal C\cong \widetilde {\mathcal C} \boxtimes \mathcal P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde {\mathcal C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an odd-dimensional modular category and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a pointed modular category of dimension 2. This reduces the classification of these categories to the classification of odd-dimensional modular categories. It follows that modular categories <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</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 F upper P d i m left-parenthesis script upper C right-parenthesis identical-to 2 left-parenthesis mod 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>P</mml:mi> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≡ </mml:mo> <mml:mn>2</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>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">FPdim(\mathcal {C})\equiv 2 \pmod 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of rank up to 46 (except for 34) are pointed. More generally, we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a weakly integral modular category and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an odd prime dividing the order of the group of invertibles that has