For Variety V, function V → K ¯ I’m reading Silverman’s Arithmetic of Elliptic Curves, and they make the comment “Since an element f ∈ K ¯ [ V ] is well-defined up to a polynomial vanishing on V, it induces a well-defines function f : V → K ¯ , where V denotes an affine algebraic variety. My question:Let’s start off by using more pedantic notation. Let f ¯ ∈ K ¯ , and let F ¯ denote this induced map V → K ¯ . How is F ¯ defined?

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For Variety V, function   V  →      K    ¯  I’m reading Silverman’s Arithmetic of Elliptic Curves, and they make the comment “Since an element   f  ∈      K    ¯    [  V  ] is well-defined up to a polynomial vanishing on V, it induces a well-defines function   f  :  V  →      K    ¯    , where V denotes an affine algebraic variety. My question:Let’s start off by using more pedantic notation. Let       f    ¯    ∈      K    ¯  , and let       F    ¯   denote this induced map   V  →      K    ¯  . How is       F    ¯   defined?

HadoHaurrysap3w · Accepted Answer

Step 1Do you understand how the ring       C    [  x  ,  y  ]      /    (      y    2    −      x    3    −  1  ) is constructed ? Its elements are functions from the elliptic curve   E  =  {  (  a  ,  b  )  ∈            C        2    ,      b    2    −      a    3    −  1  =  0  } to C in the natural way  (      ∑          i      ,      j            c          i      ,      j            x    i        y    j    )  (  a  ,  b  )  =      ∑          i      ,      j            c          i      ,      j            a    i        b    j  Step 2The main concept to investigate is that we can evaluate any polynomial of C[x,y] on E, then the ideal   (      y    2    −      x    3    −  1  ) is exactly the subset of those vanishing on E.

For Variety V, function V rightarrow overlineK

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