jsMath

# Iwasawa theory and p -adic Analogues of the BSD Conjecture

Christian Wuthrich and I are finishing a paper right now called Computations About Tate-Shafarevich Groups Using Iwasawa Theory, which combines a range of theorems of Kato, Perrin-Riou, Schneider, and others to prove the following result in the rank 1 case:

 Theorem (p-adic Iwasawa-style bound): Let E  be an elliptic curve over Q  with analytic rank 1  having good ordinary reduction at an odd prime p . Suppose that representation ÚE;p  is either surjective or reducible and that the p -adic regulator Regp(E)  is nonzero. Then ordp(#Sha(E=Q))Ôordp(the BSD conjectural order of Sha):

NOTE: I didn't mention either Iwasawa theory of p -adic L -functions in the statement of the theorem! However, there is an important refinement of this theorem at bad multiplicative and supersingular primes, which gives an explicit bound in terms of certain p-adic regulators that we will not define here.

The case of primes of additive reduction remains problematic.

 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field
 3Á47  3Á47

Note that 7 is a prime of good ordinary reduction for E :
 (True, True) (True, True)

Next we compute the 7 -adic regulator and verify that it is nonzero. Due to a recent paper of Mazur-Stein-Tate that builds on Kedlaya's algorithm, and much optimization work of David Harvey, SAGE can quite quickly compute p -adic heights to very high precision and for surprisingly large p .
 4Á7À1+3+4Á7+6Á72+3Á73+3Á74+3Á75+3Á76+77+6Á79+3Á710+6Á711+712+2Á713+4Á714+6Á715+4Á716+5Á718+719+O(720)  4Á7À1+3+4Á7+6Á72+3Á73+3Á74+3Á75+3Á76+77+6Á79+3Á710+6Á711+712+2Á713+4Á714+6Á715+4Á716+5Á718+719+O(720)

Since p=7  is a good ordinary prime and E  has rank 1 , it's not necessary to compute the 7 -adic L -function of E . Nonetheless we do that for fun (and it would have been needed if p  were supersingular or a prime of multiplicative reduction):
 7-adic L-series of Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field 7-adic L-series of Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field
 À1+3Á7+6Á72+3Á73+3Á74+6Á75+O(76)ÁT+À6Á7+5Á72+6Á74+6Á75+O(76)ÁT2+À6+4Á7+2Á73+5Á74+2Á75+O(76)ÁT3+O(T4) À1+3Á7+6Á72+3Á73+3Á74+6Á75+O(76)ÁT+À6Á7+5Á72+6Á74+6Á75+O(76)ÁT2+À6+4Á7+2Á73+5Á74+2Á75+O(76)ÁT3+O(T4)
 À1+3Á7+72+3Á74+75+3Á76+4Á77+2Á78+2Á79+O(710)ÁT+À6Á7+72+2Á73+2Á74+75+76+5Á77+78+5Á79+O(710)ÁT2+À6+4Á7+5Á72+73+6Á74+2Á75+5Á76+6Á77+78+2Á79+O(710)ÁT3+À1+2Á7+3Á72+4Á73+3Á74+4Á75+6Á76+5Á77+6Á78+O(710)ÁT4+O(T5) À1+3Á7+72+3Á74+75+3Á76+4Á77+2Á78+2Á79+O(710)ÁT+À6Á7+72+2Á73+2Á74+75+76+5Á77+78+5Á79+O(710)ÁT2+À6+4Á7+5Á72+73+6Á74+2Á75+5Á76+6Á77+78+2Á79+O(710)ÁT3+À1+2Á7+3Á72+4Á73+3Á74+4Á75+6Á76+5Á77+6Á78+O(710)ÁT4+O(T5)

ASIDE: There is a formula of Perrin-Riou that one could also use to compute L0p(E;0) when p  is good ordinary:
1Reg(E=Q)ÁÊEL0(E;1)=1Regp(E=Q)ÁL0p(E;0)(1À1Ë)2Álog(Ô(Í))
which is an equality of rational numbers.

This formula is why we didn't have to mention p -adic L -functions in the theorem above.