# Euler characteristics in the rank one case

Suppose $$E$$ is an elliptic curve over a number field with good ordinary reduction at the primes above a fixed odd prime $$p$$. We are interested in the Iwasawa theory over the cyclotomic $$\mathbb{Z}_p$$ extension under the additional assumption that $$E$$ has a point of infinite order and also that $$L(E,s)$$ has a simple zero at $$s=1$$.

Is there a definition of a modified Euler characteristic for the Selmer group over the cyclotomic extension which can be related to the derivative $$L'(E,1)$$ in the framework of the BSD conjecture? Can someone give me some precise references.

Let $$E$$ be an elliptic curve over a number field $$F$$ and $$p$$ a prime such that $$E$$ has good ordinary reduction at all places above $$p$$. Suppose we know that the dual $$X$$ of the usual Selmer group over the cyclotomic $$\mathbb{Z}_p$$-extension is torsion, e.g. if $$F=\mathbb{Q}$$.

Then we can compute the order of vanishing and the leading term of the characteristic series of $$X$$ as a power series in $$\mathbb{Q}_p[\![T]\!]$$. If the canonical $$p$$-adic height is non-degenerate, as expected, then the order of vanishing is equal to corank of the Selmer group over $$F$$, i.e. equal to the rank of $$E$$ if you believe the finiteness of Sha. The leading term formula, still under the assumption of the non-degeneracy of the $$p$$-adic height, is a $$p$$-adic BSD-type formula. This was first proven by Perrin-Riou and Schneider. It involves the $$p$$-adic regulator, Tamagawa numbers, torsion square, the order of Sha and a correction term like in the interpolation formula for the $$p$$-adic $$L$$-function at the trivial character. Of course the formula is only up to a $$p$$-adic unit as the characteristic series is only defined up to a unit. If there is an analytic $$p$$-adic $$L$$-function to compare to, the formula compares well with the expected $$p$$-adic BSD formula there.

When proving this formula, one has to compare $$X^\Gamma$$ and $$X_\Gamma$$ where $$\Gamma$$ is the Galois group of the $$\mathbb{Z}_p$$-extension. If they are finite then one can call it the $$\Gamma$$-Euler characteristic that one has to compute. But in the higher rank case, one uses instead the map $$X^\Gamma \to X \to X_{\Gamma}$$. Under the assumption that the $$p$$-adic height is non-degenerate, this map has finite kernel and cokernel. This is very similar to Tate's proof of the BSD formula in the function field case. Perrin-Riou generalise it to $$p$$-adic representations, but maybe her article "Théorie d'Iwasawa et hauteurs p-adiques (cas des variétés abéliennes)" is a good place to start.

Now to your question when the analytic rank is $$1$$. Let's say we are over $$F=\mathbb{Q}$$. Then we have a Heegner point $$P$$ and the rank is really $$1$$. One may express this formula in terms of the $$p$$-adic height of $$P$$. By the Gross-Zagier formula and its $$p$$-adic analogue one can compare the $$p$$-adic BSD formula to the usual one and hence express the leading term (which is a sort of a generalised Euler characteristic) in terms of $$L'(E,1)$$. I think it is best to look at the Bourbaki talk by Colmez http://www.numdam.org/item/SB_2002-2003__45__251_0/ on Kato's work where all of this is surveyed. It contains all the relevant references.

• @Wuthrich I had looked through some papers of Schneider and Perrin Riou on the p-adic height pairing, I wasn't able to find exactly what I was looking for in the rank 1 case. Let me see if this does get me to the right place. – user130124 Jun 17 '19 at 14:19
• I am not sure what "exactly" you are looking for. Colmez refers to "Points de Heegner et dérivées de fonctions L p-adiques" by Perrin-Riou for the p-adic Gross-Zagier formula. That part is specific to rank 1, otherwise the reference to Perrin-Riou in my answer applies to all ranks. – Chris Wuthrich Jun 17 '19 at 15:28
• @Wuthrich The formula (in the exact form I'm looking for) is stated on page 204 of "Links between Cyclotomic and $GL_2$ Iwasawa theory" by Coates, Schneider, Sujatha. It does express the modified Euler characteristic in terms of the determinant of the $p$-adic height pairing and some other expected terms. It isn't specific to rank 1. Perhaps I could have derived this by looking into the reference you cited if I read through it more carefully. – user130124 Jun 22 '19 at 9:18
• ... (I found the reference I was looking for and am satisfied). – user130124 Jun 22 '19 at 9:19
• @Wuthrich This is very related: I'm interested in applying Theorem 2' (which in fact is the formula relating the derivative with the truncated Euler characteristic) in Schneider's p-adic height pairings 2 for elliptic curves with bad reduction at p, everything that's carried out in his paper is in the good reduction case, would you happen to know if this formula is noted anywhere else where maybe it would be applicable without the good reduction hypothesis? – user130124 Sep 6 '19 at 6:36