The existence of supersingular curves of genus 4 in arbitrary characteristic
Momonari Kudo, Shushi Harashita, Hayato Senda
Abstract
Abstract We prove that there exists a supersingular nonsingular curve of genus 4 in arbitrary characteristic $$p>0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . For $$p>3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> we shall prove that the desingularization of a certain fiber product over $$\mathbf{P }^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> of two supersingular elliptic curves is supersingular.