Investigate the order of poles of a rational function at the point of infinity on an elliptic curve using a computational approach | Step-by-Step Solution

What We're Solving:

We want to analyze how a rational function behaves near the "point at infinity" on an elliptic curve, specifically looking at where the function "blows up" (has poles) and how severely it does so (the order of those poles). We're working over a field with finite characteristic greater than 3, which gives us some nice computational advantages.

The Approach:

We're going to:

• Set up our elliptic curve in a form that makes the point at infinity visible
• Use local coordinates around that special point
• Expand our rational function as a power series
• Count how many negative powers appear (that's our pole order)

The reason we do this is fundamental: understanding pole behavior tells us about the function's geometric properties and helps classify rational functions on elliptic curves.

Step-by-Step Solution:

Step 1: Set up the elliptic curve properly Start with your elliptic curve in Weierstrass form: y² = x³ + ax + b (since char > 3, we don't need the xy and y terms). To see the point at infinity clearly, we need to work in projective coordinates [X:Y:Z] where our curve becomes: Y²Z = X³ + aXZ² + bZ³

The point at infinity is [0:1:0] - this is where Z = 0.

Step 2: Choose appropriate local coordinates Near the point at infinity, we use the substitution:

• x = X/Z = 1/t (so t is our local parameter)
• y = Y/Z = s/t³

This transforms our curve equation and makes the point at infinity correspond to t = 0.

Step 3: Express your rational function in local coordinates Whatever rational function f(x,y) you're analyzing, substitute x = 1/t and y = s/t³. This will give you f as a function of t and s, where the curve constraint becomes a local equation in these variables.

Step 4: Expand as a Laurent series Near t = 0, expand f(t,s) as a power series in t: f(t,s) = a₋ₖt⁻ᵏ + a₋ₖ₊₁t⁻ᵏ⁺¹ + ... + a₋₁t⁻¹ + a₀ + a₁t + ...

Step 5: Identify the pole order The order of the pole at infinity is the largest power k such that the coefficient a₋ₖ ≠ 0. If k > 0, you have a pole of order k. If the expansion starts with non-negative powers, there's no pole.

Step 6: Use the computational advantage Since we're in characteristic > 3, our Weierstrass form is simplified, and many computations become cleaner. You can often use computer algebra systems effectively here.

The Framework:

Since this appears to be a research-oriented problem, here's how to structure your analysis:

• 1. Setup section: Define your specific elliptic curve and rational function
• 2. Coordinate transformation: Show the projective setup and local coordinates
• 3. Series expansion: Perform the actual computation
• 4. Results interpretation: What does the pole order tell you geometrically?
• 5. Conclusions: How does this fit into the broader theory of rational functions on elliptic curves?

Memory Tip:

Remember "PIT" - Projective coordinates to see infinity, Invert to get local parameters (x = 1/t), Taylor expand to find poles! The point at infinity is just like any other point once you choose the right coordinates to look at it.

The beauty of this problem is that it connects abstract algebraic geometry with concrete computation. You're essentially using calculus-like techniques (series expansions) to understand deep geometric properties. Keep thinking about WHY each step makes geometric sense, and the computation will follow naturally!

You've got this! Algebraic geometry can feel abstract, but problems like this show how we can make these beautiful concepts very concrete and computable.