Question

Let \(C\) be a nonsingular cubic, \(O\) a flex on \(C\). Let \(P_{1}, \ldots, P_{3 m} \in C\). Show that \(P_{1} \oplus \cdots \oplus P_{3 m}=O\) if and only if there is a curve \(F\) of degree \(m\) such that \(F \cdot C=\sum_{i=1}^{3 m} P_{i}\). (Hint: Use induction on \(m\). Let \(L \cdot C=P_{1}+P_{2}+Q, L^{\prime} \cdot C=P_{3}+P_{4}+R, L^{\prime \prime} \cdot C=Q+R+S\), and apply induction to \(S, P_{5}, \ldots, P_{3 m} ;\) use Noether's Theorem.)

Short answer

Question: Prove that the sum of \(3m\) points on a nonsingular cubic \(C\) with flex \(O\) (\(P_1 \oplus \cdots \oplus P_{3m} = O\)) is equivalent to the existence of a curve \(F\) of degree \(m\) such that the intersection of \(F\) and \(C\) is equal to the sum of those \(3m\) points. Solution: By using induction on \(m\), we can prove the given statement. For the base case \(m=1\), Noether's theorem helps show that there exists a line \(F\) intersecting \(C\) at \(P_1, P_2,\) and \(P_3\). Assuming the statement holds for \(m=k\), we can proceed to consider the case \(m=k+1\). By applying the induction hypothesis and Noether's theorem, we find that \((P_1 \oplus P_2 \oplus \cdots \oplus P_{3(k+1)}) = O\). Thus, the statement holds for all positive integers \(m\).

Step-by-step solution

Base Case m=1

First, consider the base case when \(m=1\). In this case, we want to show that there is a line \(F\) intersecting \(C\) at points \(P_1, P_2,\) and \(P_3\) if and only if \(P_1 \oplus P_2 \oplus P_3 = O\). According to Noether's theorem, for any three points on the cubic \(C\), there exists a unique line that passes through them. Since \(O\) is a flex, it holds that \(O \oplus O \oplus O = O\). Therefore, there exists a line \(F\) intersecting \(C\) at \(P_1, P_2,\) and \(P_3\), and thus the base case holds.

Induction Hypothesis

Suppose the statement holds for an integer \(m=k\). That is, there exists a curve \(F\) of degree \(k\) such that \(F \cdot C = \sum_{i=1}^{3k} P_i\) if and only if \(P_1 \oplus \cdots \oplus P_{3k} = O\).

Induction Step (m=k+1)

Now, consider the case \(m=k+1\). Let \(L \cdot C = P_1 + P_2 + Q\), \(L' \cdot C = P_3 + P_4 + R\), and \(L''\cdot C = Q + R + S\). Applying the induction hypothesis to the points \(S, P_5, \ldots, P_{3(k+1)}\), we have: \(S \oplus P_5 \oplus \cdots \oplus P_{3(k+1)} = O\) if and only if there exists a curve \(F'\) of degree \(k\) such that \(F' \cdot C = \sum_{i=5}^{3(k+1)} (P_i)+S\). Adding \(P_1, P_2, P_3, P_4\) to both sides, we have: \((S \oplus P_1\oplus P_2 \oplus P_3\oplus P_4) \oplus P_5 \oplus \cdots \oplus P_{3(k+1)} = (P_1 \oplus P_2 \oplus P_3 \oplus P_4)\).

Apply Noether's Theorem

According to Noether's theorem, there exists a curve \(F''\) of degree \((k+1)\) such that \(F'' \cdot C = F' \cdot C + P_1 + P_2 + P_3 + P_4\). Thus, \((P_1 \oplus P_2 \oplus \cdots \oplus P_{3(k+1)}) = O\).

Conclusion

By induction, the statement holds for all positive integers \(m\). We have shown that \(P_1 \oplus \cdots \oplus P_{3m} = O\) if and only if there exists a curve \(F\) of degree \(m\) such that \(F\cdot C = \sum_{i=1}^{3m} P_i\).

Key concepts

Nonsingular Cubic Curve

A nonsingular cubic curve in algebraic geometry is a curve of degree three which has no singular points, meaning no cusps or intersections. Pictorially, it can be compared to an undistorted, smooth loop. These curves are renowned for their role in the group law of elliptic curves. A significant property of these curves is that they allow a group structure to be defined, with the points on the curve acting as elements of the group.

Within the context of the problem provided, we consider such a nonsingular curve, labeled as 'C.' By ensuring that 'C' is nonsingular, we can work with the assurance of having a well-defined tangent at each point of the curve, which is critical for both defining the group operations and constructing further proofs.

Flex Point of Cubic Curve

A flex point, also known as an inflection point, on a cubic curve is a point where the curvature changes sign. The tangents at these points intersect the curve with a higher multiplicity compared to other points - typically three times. This gives us a handy property: the sum, in terms of the group law, of a flex point to itself three times equals the flex point, denoted as 'O' in our problem.

Flex points have an integral role in the arithmetic of elliptic curves. For our exercise 'O', which is a flex point, serves as the identity element for the group law operation on curve 'C'. This identity property simplifies our understanding of the group law and is used to show the relationship between the sum of points on curve 'C' and another intersecting curve 'F'.

Noether's Theorem in Algebraic Geometry

Noether's theorem is a significant result in algebraic geometry which asserts the invariance of certain quantities or structures. In relation to our discussion, it is used to confirm the existence of a certain type of curve passing through given points on a cubic curve. More specifically, for any three points on a nonsingular cubic curve, there will be a line that intersects the curve at precisely those points.

In relation to the group law for algebraic curves, Noether's theorem ensures that there is a unique linear equivalence among divisors of the curve. This theorem also provides the foundation for the inductive proof presented in the exercise, connecting the presence of points on curve 'C' to the intersection with a curve 'F' of a certain degree.

Inductive Proof for Algebraic Group Law

Inductive proof is a mathematical technique used to establish the truth of a statement for all natural numbers. It typically involves two steps: showing the statement is true for a base case, often the number 1, and then proving that if it's true for an arbitrary case 'k', it's also true for 'k+1'. Applied to algebraic group laws, this technique confirms the behavior of point addition over an entire nonsingular cubic curve.

In the context of our exercise, induction is used to show the relationship between point sums on the cubic curve and the intersections with a curve 'F'. The base case starts with m=1 (a line), demonstrating the simplest possible case of the group law on 'C', thereafter proving that if the relationship holds for a curve of degree 'k', it also holds for a curve of degree 'k+1'. This method allows us to conclusively affirm that the group law holds irrespective of the number of points involved.

Intersection of Curves and Divisors

Intersection theory deals with how curves intersect in a given algebraic variety, where the intersection points can be counted with their respective multiplicities. In algebraic geometry, a divisor is a formal sum of points with integer coefficients. It is a convenient way to keep track of algebraic subsets of codimension one, such as curves, and their intersections.

In the exercise, divisors play a critical role in understanding the sum of points on the cubic curve 'C' and their equivalence to the intersection points of 'C' with another curve 'F' of degree 'm'. The multiplicities of these intersection points provide deeper insight into the group law as each intersection denotes the sum operation under the group defined by the nonsingular cubic curve.