Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms" (Asterisque 295, 2004), gives rise to an element $\mathbf{z}\_{\rm Kato}$ lying in the Iwasawa cohomology $ H^1\_{\mathrm{Iw}}(\mathbb{Q}\_p, T)[\frac{1}{p}]$. In Theorem 12.5(4) of the paper, Kato shows that if the image of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ in $\mathrm{Aut}(T)$ contains $\mathrm{SL}\_2(\mathbb{Z}\_p)$, then in fact $\mathbf{z}\_{\rm Kato} \in H^1\_{\mathrm{Iw}}(\mathbb{Q}\_p, T)$.


Is it known if there are weaker conditions that are sufficient to ensure that $\mathbf{z}\_{\rm Kato}$ has this integrality property? Are there examples where it is genuinely non-integral, or is it conjectured that it should always be so?


I would be the last person to claim I understand Kato's argument, but it looks to me as if he only actually uses the weaker statement that the mod $p$ Galois representation $T/pT$ is irreducible. I'd be interested to know if this weaker condition is indeed sufficient, and whether anything is known in this direction if the weaker condition doesn't hold (i.e. if $E$ admits a $p$-isogeny). 


(EDIT: Added more detail and references.)



Are Kato's zeta elements integral?
There are two issues. Let $H=H^1\_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V\_{\mathbb{Z}\_p}(f)(1)$ and $f$ is the modular form associated to the isogeny class of $E$.


(1) What is $T$ ?


$T$ will correspond to the lattice $\Lambda$ in $\mathbb{C}$ generated by all modular symbols. This is because $V\_{2,\mathbb{Z}}(f)$ is the image of $V\_{2,\mathbb{Z}} (Y\_1(N)) = H\_1 \bigl( X\_1(N)(\mathbb{C}),\{\text{cusps}\},\mathbb{Z}\bigr)$ inside $H\_1(E(\mathbb{C}),\mathbb{Z})$. The lattice $\Lambda$ will contain the lattice $\Lambda\_0$ of the $X\_0$-optimal (strong Weil) curve $E\_0$, but it may be strictly larger if $E\_0$ has rational torsion points, e.g. for 11a. Note also that $\Lambda$ need not be the lattice of an elliptic curve in the isogeny class, but rather $\tfrac{1}{2}\Lambda\_E$ for some elliptic curve $E$. in case $E\_0[2] \subset E\_0(\mathbb{Q})$, e.g. for 17a. But can only happen for $p=2$ and for $p$ of additive reduction. So if $p$ is a prime of odd semi-stable reduction, then $T$ is the Tate-module of an elliptic curve $E\_\*$ which is an étale quotient of $E\_0$. So let $E$ be this curve for the rest of the answer.


(2) Are Kato's elements integral in $H$ ?


There are two kinds of them. The $z\_{\gamma}$ and the ${c,d}$ $z\_{m}$. (sorry I don't seem to be able to produce indices before the symbol in MathJax) The latter are in $H$, see 8.1 of Kato, but they depend on the choices of $c$ and $d$. They are useful for bounding the Selmer group as, for a fixed $c$ and $d$ they form an Euler system.


The $z\_{\gamma}$ instead is linked to the $p$-adic $L$-function and they are independent of the choices. They are obtained by dividing by $\mu(c,d)$, page 229 of Kato. So they need not be integral anymore. The appendix A in Delbourgo's book "Elliptic curves and big Galois representations" discusses this in detail. Kato shows that they are in $H\otimes \mathbb{Q}\_p$


Kato shows that $z\_{\gamma}$ is integral if $H$ is a free $\Lambda$-module of rank 1, e.g. as shown in 12.4.(3) if $T/pT$ is irreducible. In fact it is not hard to show that $H$ is free also if $E(\mathbb{Q})[p]$ is trivial. 


Now if the curve admits an isogeny of degree $p$, one can show that for all curves $A$ in the isogeny class $H^1\_{\text{Iw}}(\mathbb{Q},T\_p A)$ is a free $\Lambda$-module of rank $1$, except for at most a single one of them (I mean up to non-$p$-isogenies of course). This exception - if present - will always be the minimal curve $E\_{\text{min}}$ in the class. Moreover if $E\_{\text{min}}$ is does not have a free $H^1\_{\text{Iw}}$, then there is an embedding of it into $\Lambda$ with image equal to the maximal ideal of $\Lambda$. One can now conclude from the interpolation property of the $p$-adic $L$-function that even if $E\_\*=E\_{\text{min}}$ then $z\_{\gamma}$ will be in $H$.


Using this one can prove that the $p$-adic $L$-function is integral and that $\mu\geq 0$, but it would not say anything about Greenberg's $\mu=0$ conjecture. Furthermore one gets a proof of the divisibility in the main conjecture as in 12.5.(3), if the $T/pT$ is reducible. However this conclusion can not be extended at present to all odd semi-stable primes, because there may be primes for which the Galois representation is not surjective, yet $T/pT$ is irreducible; because the Euler system method requires and element $\binom{1\ 1}{0\ 1}$ in the image of the Galois representation, see Hyp($K\_{\infty},T)$ in Rubin's book. So the integrality won't help, yet. 


In summary, all zeta-elements of Kato are integral with respect to $T$ in the case of an elliptic curve. For Kato's divisibility on the other hand, the surjectivity of the representation to $\mathrm{GL}\_(\mathbb{Z}\_p)$ or its reducibility is still needed.


(edits: quite a few in the whole answer above, now that I konw the full answer to the question)


