# Why does birational equivalence preserve group law in elliptic curves?

In the first chapter of "Rational points on Elliptic curves", Silverman, page 24 writes

The transformations we used to put the curve in normalized form do not map straight lines to straight lines. Since we defined the group law on our curve using lines connecting points, it is not at all clear that our transformation preserves the structure of the group. (That is, is our trans- transformation a homomorphism?) It is, but that is not at all obvious. The point is that our description of addition of points on the curve is not a good one, because it seems to depend on the way the curve is embedded in the plane. But in fact the addition law is an intrinsic operation which can be described on the curve and is invariant under birational transformation. This follows from basic facts about algebraic curves, but is not so easy (virtually impossible?) to prove simply by manipulating the explicit equations.

This seems really important and really surprising (at least to me)! Is this addressed later on in the book? What would be a good resource to get a at sketch of that proof, or at least to get a sense of why it works.

• If you're willing to work in the complex case, to understand the gist. Here's what you can do: prove that a hlomoprhic map between 1-dimensionnal complex tori is automatically a group homomorphism as long as it preserves the unit in the group. Between complex curve, a rational map is automatically a morphism, so this is sufficient to prove the invariance by birational transform. – Ahr Jan 15 '18 at 11:18

## 1 Answer

Reference: Hartshorne's Algebraic Geometry, Chapter II, Example 6.10.2. Hartshorne shows that the points of a nonsingular cubic curve with a distinguished point naturally correspond to the elements of the group of classes of divisors of degree $0$ on the curve modulo principal divisors. The operation of addition of divisors on a curve and the notion of a principal divisor do not depend on the embedding of the curve in projective space, therefore the group operation on the set of points is preserved by birational equivalence.

• Thank you. Do you think I can consume Chapter II as a standalone or do you recommend reading the whole thing before going back to Silverman's? – Arthur B. Jan 15 '18 at 14:39
• Some people recommend reading the Chapters II and III of Hartshorne (and doing exercises) to those who learn algebraic geometry. I know almost nothing about elliptic curves, so I cannot give an advice to you. – danneks Jan 15 '18 at 15:37
• Also, I learned mostly from Ravi Vakil's Math 216 notes. – danneks Jan 15 '18 at 16:10