Ranks

1. Ranks June 3, 2009

4. Which integers m are the sum of two rational cubes?

5. Which integers m are the sum of two rational cubes?

6. 346 is the sum of two rational cubes

7. Selmer’s table of A for which has infinitely many solutions

8. Conjecture from random matrix theory:

10. When p is 1 mod 3 there are three solutions to with a congruent to 2 mod 3. One of these has b divisible by 3. The corresponding a is defined to be p=7, 4p=28

11. Gauss showed that the number of solutions modulo p to is given by

12. Conjecture (Watkins): Fix a prime p. Then

13. For example:

15. Elliptic curves Let E be an elliptic curve with equation

17. Usually, when the root number is +1 the rank is 0 and when the root number is -1 the rank is 1

19. By a conjecture of Goldfeld, we expect that in the family of quadratic twists the rank will be 0 almost all of the time that the root number is +1 and that the rank will be 1 almost all of the time that the root number is -1. Question: How often do we get rank 2 or 3 in the family of quadratic twists of a fixed elliptic curve?

20. By random matrix theory we expect rank two curves to occur in the family of quadratic twists of E for about

21. Example:

22. Alternatively

24. Formula for central value: Gross found:

26. The quadratic twists form a family that seems to have orthogonal symmetry. Since we are restricting attention to L-functions with a + sign in the functional equation, we should model the distribution of values of these L-functions by the distribution of values of characteristic polynomials from SO(2N). In particular, the moments should match up:

28. Keating and Snaith formula for s’th moment of orthogonal characteristic polynomials: Probability density function: For small and large N,

31. Same as last slide but with more than 2000 curves. The theta series were supplied by Gonzalo Tornaria and Fernando Rodriguez-Villegas.

33. How are rank 2 twists distributed in arithmetic progressions? Conjecture: (C, Keating, Rubinstein, Snaith)

35. A second order approximation

40. What about rank 3 in the family of quadratic twists?

41. Nina Snaith’s derivative calculation

42. Note: The rightmost pole is at s=-3/2.

43. Suggestion for the frequency of rank 3 vanishing Might be plausible based on Elkies data for rank 3 curves among twists of the congruent number curve. RMT suggests

44. Elkies data about rank 3 twists of the congruent number curve, sorted by Watkins The first column is the prime. The second column is the number of rank 3’s in square residue classes. The third column is the number of rank 3’s in non-square classes. The fourth column is the ratio of columns two and three. The last column is the RMT prediction.

45. Saturday night conjecture

49. Quadratic twists of weight 4 and 6 modular forms.

50. 3 1.18 1.11 5, 0.55 0.59 11, 1.06 1.15 13, 0.86 0.84 17, 0.84 0.76 19, 1.35 1.53 23, 0.92 0.87 29, 1.14 1.22 31, 0.99 1.05 37, 1.19 1.17 41, 0.90 0.82 43, 0.93 0.90 47, 0.87 0.76 53, 1.06 1.06 59, 0.79 0.75 61, 1.14 1.15 67, 0.95 0.94 71, 1.16 1.17 73, 1.14 1.08 79, 0.93 0.93 83, 1.21 1.18 89, 0.91 0.87 97, 0.97 0.98 Vanishings of twists of the level 7 weight 4 cusp form. There are 1155 vanishings out of 13298378 twists up to d=100,000,000 The first column is the prime, the second is the random matrix prediction; the last is the data. The RMT prediction is