U_{p} (\frac{1}{j})

William Stein

December 2003

U_{5} ( \frac{1}{j} ) ≡ 0 (mod 5)
U_{5} ( \frac{1}{j} ) ≡ 940 ( \frac{1}{j} ) (mod 5^{5})
U_{5} ( \frac{1}{j} ) ≡ 10315 ( \frac{1}{j} ) + 3125 ( \frac{1}{j} )^{2} (mod 5^{6})
U_{5} ( \frac{1}{j} ) ≡ 744690 ( \frac{1}{j} ) + 1596875 ( \frac{1}{j} )^{2} + 3781250 ( \frac{1}{j} )^{3} (mod 5^{10})
U_{5} ( \frac{1}{j} ) ≡ 20275940 ( \frac{1}{j} ) + 21128125 ( \frac{1}{j} )^{2} + 13546875 ( \frac{1}{j} )^{3} + 9765625 ( \frac{1}{j} )^{4} (mod 5^{11})
U_{5} ( \frac{1}{j} ) ≡ 18819104065 ( \frac{1}{j} ) + 1730112500 ( \frac{1}{j} )^{2} + 29066281250 ( \frac{1}{j} )^{3} + 1523437500 ( \frac{1}{j} )^{4} + 20800781250 ( \frac{1}{j} )^{5} (mod 5^{15})
U_{5} ( \frac{1}{j} ) ≡ 49336682190 ( \frac{1}{j} ) + 428976206250 ( \frac{1}{j} )^{2} + 273206906250 ( \frac{1}{j} )^{3} + 306699218750 ( \frac{1}{j} )^{4} +

539599609375 ( \frac{1}{j} )^{5} + 640869140625 ( \frac{1}{j} )^{6} (mod 5^{17})

U_{5} ( \frac{1}{j} ) ≡ 49336682190 ( \frac{1}{j} ) + 26368917612500 ( \frac{1}{j} )^{2} + 90300062375000 ( \frac{1}{j} )^{3} + 42268369140625 ( \frac{1}{j} )^{4} +

33345996093750 ( \frac{1}{j} )^{5} + 83801269531250 ( \frac{1}{j} )^{6} + 77056884765625 ( \frac{1}{j} )^{7} + (mod 5^{20})

U_{5} ( \frac{1}{j} ) ≡ 49336682190 ( \frac{1}{j} ) + 1266145528940625 ( \frac{1}{j} )^{2} + 1711546400265625 ( \frac{1}{j} )^{3} + 900575253906250 ( \frac{1}{j} )^{4} +

1940694628906250 ( \frac{1}{j} )^{5} + 2372619628906250 ( \frac{1}{j} )^{6} + 363159179687500 ( \frac{1}{j} )^{7} + 2002716064453125 ( \frac{1}{j} )^{8} (mod 5^{22})

The 5-adic valuations of the coefficients of the power series U_{5}(1/j) are

1, 5, 6, 10, 11, 15, 17, 20, 22, 25, 26, 30, 31, 35, 36, 40, 43, 46, ….

On the other hand, if p = 7, then the 7-adic valuations of the coefficients of U_{7}(1/j), written in terms of 1/j are

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ….

Note that 0 is not a supersingular j-invariant in characteristic 7.

When p = 11, the invariant 0 is supersingular, but 1 is also supersingular. The 11-adic valuations of the coefficients of U_{11}(1/j), written in terms of 1/j, are

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ….