Involution On elliptic curve

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In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ under the the group operation on $E$?



One way is to use the explicit formula in terms of x,y for the inverse, I suppose.



But in the book I’m reading (Hida’s Geometric Modular Forms) the group structure on $E$ is not defined via the usual secant line process,but via the isomorphism $E$ with $Pic^0$ sending $P$ to $I(P)^-1 otimes I(O)$ where $I(P)$ is the ideal sheaf of the point $P$ and $O$ is the origin. Somehow it seems, from the highlighted part in the excerpt below, we need to use something about how the group operation interacts with the a non-vanishing 1-form - we’ve seen earlier that all such are invariant under pullback via addition/translation by given point of $E$
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algebraic-geometry elliptic-curves arithmetic-geometry




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    In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ under the the group operation on $E$?



    One way is to use the explicit formula in terms of x,y for the inverse, I suppose.



    But in the book I’m reading (Hida’s Geometric Modular Forms) the group structure on $E$ is not defined via the usual secant line process,but via the isomorphism $E$ with $Pic^0$ sending $P$ to $I(P)^-1 otimes I(O)$ where $I(P)$ is the ideal sheaf of the point $P$ and $O$ is the origin. Somehow it seems, from the highlighted part in the excerpt below, we need to use something about how the group operation interacts with the a non-vanishing 1-form - we’ve seen earlier that all such are invariant under pullback via addition/translation by given point of $E$
    enter image description here




    algebraic-geometry elliptic-curves arithmetic-geometry




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      In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ under the the group operation on $E$?



      One way is to use the explicit formula in terms of x,y for the inverse, I suppose.



      But in the book I’m reading (Hida’s Geometric Modular Forms) the group structure on $E$ is not defined via the usual secant line process,but via the isomorphism $E$ with $Pic^0$ sending $P$ to $I(P)^-1 otimes I(O)$ where $I(P)$ is the ideal sheaf of the point $P$ and $O$ is the origin. Somehow it seems, from the highlighted part in the excerpt below, we need to use something about how the group operation interacts with the a non-vanishing 1-form - we’ve seen earlier that all such are invariant under pullback via addition/translation by given point of $E$
      enter image description here




      algebraic-geometry elliptic-curves arithmetic-geometry




      share|cite|improve this question











      In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ under the the group operation on $E$?



      One way is to use the explicit formula in terms of x,y for the inverse, I suppose.



      But in the book I’m reading (Hida’s Geometric Modular Forms) the group structure on $E$ is not defined via the usual secant line process,but via the isomorphism $E$ with $Pic^0$ sending $P$ to $I(P)^-1 otimes I(O)$ where $I(P)$ is the ideal sheaf of the point $P$ and $O$ is the origin. Somehow it seems, from the highlighted part in the excerpt below, we need to use something about how the group operation interacts with the a non-vanishing 1-form - we’ve seen earlier that all such are invariant under pullback via addition/translation by given point of $E$
      enter image description here





      algebraic-geometry elliptic-curves arithmetic-geometry





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