Elliptic Curves and Modular Forms, a $1,000,000 Mystery
Episode #9 of the course Great math problems for the 21st-century mind by John Robin
Recall that I told you that for the curve y2 = x3 − x, limited to the number field p = 3 (i.e., the numbers 0, 1, and 2 only for x and y), we had only the solutions (0,0), (1,0), and (2,0). We called the number of solutions Np and said in this case, that N3 = 3.
Now, there is another number we are interested in calculating. We want to measure the difference p − Np when counting the number of solutions. For instance, here, we would have 3 − 3 = 0. We call this number the p-defect (also called the trace of Frobenius) and label it ap. So here, we write a3 = 0.
For p = 5, there are seven points: (0,0), (1,0), (2,1), (2,4), (3,2), (3,3), (4,0). Remember, for field size 5, we start at zero when we hit 5, so for example, with x = 3, we get 33 − 3 = 27 − 3 = 24 = 4, and for y = 3, we get 32 = 9 = 4. So, you can see here how this verifies that (3,3) is a solution. The same process will show the others are all solutions too.
So, for p = 5, we say the p-defect a5 = 5 − 7 = −2.
For p = 7, we have seven solutions, so a7 = 0.
For p = 11, we have eleven solutions, so a11 = 0.
For p = 13, we have seven solutions, so a13 = 6.
We can go on and on through all the prime numbers.
Now you may recall from Lesson 3 when I talked about modularity and how one can associate any non-singular elliptic curve with a polynomial of the form:
a1q + a2q2 + a3q3 + …
where a1, a2, a3, and so on, are the number of solutions in fields of increasing size. The ap values we calculate over prime-sized number fields correspond (with some exceptions that are nuanced and beyond the scope of this course) to the specific coefficients in this polynomial.
For our friend y2 = x3 − x, its modular form is an associated polynomial that has the form:
Θ = a1q + a2q2 + (0)q3 + a4q4 + (−2)q5 + …
where we have the wonderful property that, for almost all primes, p, the coefficient in the pth power of Θ matches ap from our elliptic curve.
What mathematicians have observed is that if an elliptic curve has rational points, the associated modular form takes on a particular pattern, seen only when computed for curves with known rational solutions. Specifically, a function that uses the p-defect, known as an L-function, is zero whenever an elliptic curve has infinitely many rational points, and nonzero otherwise.
Peter Swinnerton-Dyer, in the 1960s, used the EDSAC-2 computer to calculate these patterns and found that, just like the Riemann Hypothesis with its zeros of the form 1/2 + bi, elliptic curves with rational points consistently had this pattern. His subsequent analysis with mathematician Bryan Birch led them to conjecture that when an elliptic curve has rational solutions, it will always take on this form.
To date, no one can prove it to be true, and this is the proof on which the Clay Institute of Mathematics has placed a $1,000,000 prize. This proof is very important to number theorists. It means they will finally be able to rest and be reassured about many assumptions made about other results that assume the Birch and Swinnerton-Dyer Conjecture true (like Tunnell’s Theorem, which allows us to speculate on the complete set of congruent numbers).
What ground we’ve covered! I hope I’ve made you curious about the big mysteries that drive mathematicians forward. Tomorrow, I will close this course with another of my favorites—though it’s less a problem and more a matter of deepening the awe and wonder of mathematics.
If you are curious about Birch and Swinnerton-Dyer Conjecture and want to investigate it as a gateway to push your own understanding of mathematics and possibly discover something that might help in the endeavor for a proof, I recommend that you start with this great article.
Elliptic Curves and Modular Forms by Neal Koblitz
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