An Elementary Criterion (Thursday)

1. ($<1$ hour) - Student presentations: proof of FLT for exponent 4 and no square triangles.
2. (10 minutes) - break
3. (30 minutes) - statement of Tunnell's criterion. Let
   $$\Theta(q) = 1 + 2\sum_{m\geq 1} q^{m^2} = 1 + 2q + 2q^{4} + 2q^{9} + \cdots.$$
   Let
   $$f_1 = \Theta(q)\cdot (2\Theta(q^{32}) - \Theta(q^8))\cdot \Theta(q^2)$$
   and
   $$f_2 = \Theta(q) \cdot (2\Theta(q^{32}) - \Theta(q^8))\cdot \Theta(q^4).$$
   Theorem 7.1 (Waldspurger, Tunnell)   Suppose $n$ is a squarefree integer. If $n$ is odd then $L(E_n,1)=0$ if and only if the $n$ th coefficient of $f_1$ is 0 . If $n$ is even then $L(E_n,1)=0$ if and only if the $\frac{n}{2}$ th coefficient of $f_2$ is 0 .
   This is a very deep theorem. It allows us to determine whether or not $L(E_n,1)=0$ . The Birch and Swinnerton-Dyer conjecture (which is not a theorem) then asserts that $L(E_n,1)=0$ if and only if $n$ is a congruent number. Thus once enough of the Birch and Swinnerton-Dyer conjecture is proved, we'll have an elementary way to decide whether or not a (squarefree) integer $n$ is a congruent number. Namely, if $n$ is odd then (conjecturally) $n$ is a congruent number if and only if
   $$\char93 \left\{x,y,z\in\mathbb{Z} : 2x^2 + y^2 + 32z^2 = n\rig... ...ac{1}{2} \char93 \left\{x,y,z\in\mathbb{Z} : 2x^2 + y^2 + 8z^2 = n\right\}.$$
   Simiarly, if $n$ is even then (conjecturally) $n$ is a congruent number if and only if
   $$\char93 \left\{x,y,z\in\mathbb{Z} : 4x^2 + y^2 + 32z^2 = \frac... ...ac{1}{2} \char93 \left\{x,y,z\in\mathbb{Z} : 4x^2 + y^2 + 8z^2 = n\right\}.$$

4. (30 minutes) - exercises with Tunnell's criterion.
   1. Come up with a method to explicitly list each of the above sets.

   2. Prove that the elementary criterion (involving cardinality of sets) implies that none of $n=1,2,3,4$ are congruent numbers.

   3. Prove that the elementary criterion (involving cardinality of sets) implies that $n=5,6,7$ are congruent numbers. Then find a right triangle with area $5$ .

   4. Verify with SAGE that Theorem 7.1 (appears to) hold for $n<20$ . Hint: The command

      f1, f2 = tunnell_forms(30)
      

      computes the $f_1$ and $f_2$ defined above, and e.g., f1[3] returns the coefficient of $q^3$ .

   5. Question (Nathan Ryan): Is it possible to decide whether or not a prime number $p$ is a (conjectural) congruent number in time polynomial in the number of digits of $p$ ? I.e., if $p$ has a hundred digits is there any hope we could tell whether or not $p$ is (conjecturally) a congruent number i a reasonable amount of time? [[WARNING: This is an unsolved problem, as far as I know.]]