Skip to main content
Notice removed Draw attention by CommunityBot
Bounty Ended with no winning answer by CommunityBot
Notice added Draw attention by Dimitri Koshelev
Bounty Started worth 50 reputation by Dimitri Koshelev
added 4 characters in body
Source Link

Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve $$ y^2 = x^3 - 1\qquad (y^2 = x^4 - 1) $$ over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that $$p \equiv 2\ (\mathrm{mod}\ 3)\qquad (p \equiv 3\ (\mathrm{mod}\ 4)). $$ In Katsura's article "Generalized Kummer surfaces and their unirationality in characteristic p" on 30 page it is stated that over the algebraic closure $\overline{\mathbb{F}_{p^2}}$ the Kummer surface $K = \mathrm{Kum}(E_1\times E_2)$ is birationally isomorphic to the elliptic surface $$ y^2 = x^3 - t^4(t-1)^4\qquad (y^2 = x^3 - t^3(t-1)^3x). $$ Is this true over $\mathbb{F}_{p^2}$? How can I see this elliptic fibration on $K$? These particular questions are very important to answer the global one Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve $$ y^2 = x^3 - 1\qquad (y^2 = x^4 - 1) $$ over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that $$p \equiv 2\ (\mathrm{mod}\ 3)\qquad (p \equiv 3\ (\mathrm{mod}\ 4)). $$ In Katsura's article "Generalized Kummer surfaces and their unirationality in characteristic p" on 30 page it is stated that over the algebraic closure $\overline{\mathbb{F}_{p^2}}$ the Kummer surface $K = \mathrm{Kum}(E_1\times E_2)$ is birationally isomorphic to the elliptic surface $$ y^2 = x^3 - t^4(t-1)^4\qquad (y^2 = x^3 - t^3(t-1)^3x). $$ Is this true over $\mathbb{F}_{p^2}$? How can I see this elliptic fibration on $K$? These particular questions are very important to answer global one Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve $$ y^2 = x^3 - 1\qquad (y^2 = x^4 - 1) $$ over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that $$p \equiv 2\ (\mathrm{mod}\ 3)\qquad (p \equiv 3\ (\mathrm{mod}\ 4)). $$ In Katsura's article "Generalized Kummer surfaces and their unirationality in characteristic p" on 30 page it is stated that over the algebraic closure $\overline{\mathbb{F}_{p^2}}$ the Kummer surface $K = \mathrm{Kum}(E_1\times E_2)$ is birationally isomorphic to the elliptic surface $$ y^2 = x^3 - t^4(t-1)^4\qquad (y^2 = x^3 - t^3(t-1)^3x). $$ Is this true over $\mathbb{F}_{p^2}$? How can I see this elliptic fibration on $K$? These particular questions are very important to answer the global one Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Source Link

The specific elliptic fibration on the Kummer surface of the superspecial abelian surface

Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve $$ y^2 = x^3 - 1\qquad (y^2 = x^4 - 1) $$ over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that $$p \equiv 2\ (\mathrm{mod}\ 3)\qquad (p \equiv 3\ (\mathrm{mod}\ 4)). $$ In Katsura's article "Generalized Kummer surfaces and their unirationality in characteristic p" on 30 page it is stated that over the algebraic closure $\overline{\mathbb{F}_{p^2}}$ the Kummer surface $K = \mathrm{Kum}(E_1\times E_2)$ is birationally isomorphic to the elliptic surface $$ y^2 = x^3 - t^4(t-1)^4\qquad (y^2 = x^3 - t^3(t-1)^3x). $$ Is this true over $\mathbb{F}_{p^2}$? How can I see this elliptic fibration on $K$? These particular questions are very important to answer global one Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?