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19

A prime $\ell$ splits in $K_p$ for $p$ odd if and only if $$\ell^{p-1} \equiv 1 \mod p^2.$$ We can express $\lim_{s \to 1} (s-1) \zeta_{K_n(s)}$ as the product of $\frac{1}{ 1- |\mathfrak a|^{-1} }$ over primes $\mathfrak a$ of $K_n$, regualarized by dividing by the product of $\frac{1}{1-\ell}$ over primes $\ell$ of $\mathbb Q$. From this perspective, large ...

12

Let me explain a bit more what that footnote was supposed to mean. As I'm sure you know, an Euler system for a Galois representation $V$ over a number field $K$ consists of a bunch of classes in $H^1(L, V)$, as $L$ varies over a suitable class of abelian extensions of $K$. If we're willing to temporarily forget about integrality, we can project to ...

11

Given the brevity of the question, I am not sure about the precise setup, so let me assume that you are looking at some $\mathbf{Z}_p$-extension and that you assume (or know in the situation at hand) that the appearing Iwasawa modules are torsion, so that the $\lambda$-invariant is defined in the first place. For an Iwasawa module $X$, this invariant is just ...

9

Suppose $p>2$ and that $E$ has good reduction. If the reduction $\tilde E(\mathbb{F}_p)$ has no $p$-torsion then there is no $p$-torsion in $E(\mathbb{Q}_p)$. Otherwise look at the exact sequence $$0\to E(\mathbb{Q}_p)[p]\to \tilde E(\mathbb{F}_p)[p]\to \hat E(p\mathbb{Z}_p)/p\hat E(p\mathbb{Z}_p).$$ Here $\hat E$ is the formal group. So to check if a $p$...

8

The main conjecture is a theorem if the image of the mod $p$ Galois representation of E is the whole of $GL_2(\mathbf{F}_p)$. The full statement of the conjecture, which implies what you wrote about lambda and mu invariants but is quite a bit stronger, is the claim that the p-adic L-function generates the characteristic ideal of the Selmer group. This was ...

8

EDIT: There was a problem with my original answer. Details below the bottom line. If you have any elliptic curve $E/{\mathbb Q}$ with a point of order 3, then we have an exact sequence of Galois modules: $$0 \to {\mathbb Z}/3{\mathbb Z} \to E[3] \to \mu_3 \to 0.$$ Here the ${\mathbb Z}/3{\mathbb Z}$ term arises from the point of order 3 defined over ${\... 7 1) 91b3 is an example with$\mu=2$at$p=3$. Recall that the$\mu$-invariant (with respect to a prime$p$) only changes when there is an isogeny of degree$p$. More precisely it changes just be the quotient of the real Néron periods. Hence we are looking here for curves with cyclic isogenies of degree 9 defined over$\mathbb{Q}$whose kernel is in$E(\mathbb{...

6

It took me a while to realise that this is an interesting question. The formulation above makes it sound like a computational problem for a specific curve, so let me first reformulate it: Let $E/k$ be an elliptic curve over a number field $k$ and let $\varphi:E \to E'$ be a cyclic isogeny of degree $p^2$ for some prime $p$. To determine how the $\mu$-...

6

As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in Invent. Math. He proved that for each abelian number field $K/\mathbb{Q}$ and every pair $\ell\neq p$ of odd primes (I am not sure at what happens with the prime $2$), the $\ell$-part ...

6

There is unique division with remainder by a monic polynomial in $O[X]$, where $O$ is any commutative ring. When $O$ is a $p$-adically complete ring, the Weierstrass division theorem tells us there is unique division with remainder by a polynomial in $O[[X]]$ that is distinguished: monic with lower degree coefficients in the maximal ideal of $O$. Therefore, ...

5

There is an answer which follows from Kolyvagin's theory of Euler Systems, and can be found in Theorem 4.4 of K. Rubin, Kolyvagin's System of Gauss Sums, in Arithmetic Algebraic Geometry, van der Geer, Oort, Steenbrink (eds), Birkhäuser, Progress in Mathematics 89, 1991. The result is (in Rubin's words) "implicitly contained" in Kolyvagin's work and says ...

5

Yes, the main conjecture is isogeny-invariant. See here: B. Perrin-Riou, Variation de la fonction $L$ $p$-adique par isogénie, Algebraic number theory, Adv. Stud. Pure Math. 17 (1989), pp. 347-358.

5

Yes, the corank conjecture is a theorem for elliptic curves over $\mathbb{Q}$. The key to the proof is the following: Theorem (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel_p^0(E / \mathbb{Q}_\infty) = \operatorname{ker}\Big(Sel_p(E / \mathbb{Q}_\infty) \to H^1(\mathbb{Q}_{p, \infty}, E[p^\infty])\Big)$ is cotorsion. One of the ...

4

1) Iwasawa theory, as practiced by K.Iwasawa, is concerned with $\mathbb Z_{p}$-extensions. There is only one $\mathbb Z_{p}$-extension of $\mathbb Q$. Over more generally number fields, and in more general context, it is emphatically not true that only the cyclotomic extension is considered. The reason why we consider the finite extensions of $\mathbb Q$ ...

4

The full main conjecture is invariant under isogenies defined over $\mathbb{Q}$, not just the statement about $\mu$ and $\lambda$-invariants. The only thing that changes on the analytic side is the period: Recall the $p$-adic $L$-function interpolates values of the form Gaussum*L-value/Period. The complex $L$-function is invariant under isogeny, so only ...

4

You can find many examples like this using Rob Pollack's tables of elliptic curve Iwasawa invariants: http://math.bu.edu/people/rpollack/Data/curves1-5000 Just search the table for a curve with $\mu=1$ at $p=3$ (this is relatively rare), and then check whether it has rational torsion at LMFDB. Here's what appears to be the example with smallest ...

4

Yes. If $E$ has potentially good reduction, this is due to H. Imai, Proc Japan Adac Math Sci 51 (1975). A non-standard proof is Theorem A.2.8 in Coates-Sujatha's "Galois cohomology of elliptic curves"

4

(I was hoping somebody else would answer this, because function fields are not really my area and I hoped I would learn something from the answer; but nobody seems to be biting, so...) Iwasawa theory over function fields definitely exists, and in many ways it's easier than number-field Iwasawa theory -- there are more nice tools available, such as the ...

4

For $n$ large enough the Galois group of $F_n/F_{n-1}$ identifies with the kernel $G$ of $\operatorname{GL}_2(\mathbb{Z}/p^{n+1}\mathbb{Z}) \to \operatorname{GL}_2(\mathbb{Z}/p^{n}\mathbb{Z})$. Let $P$ and $Q$ be a basis of $E_{p^{n+1}}$. The norm of $P$ is written as a sum over $a$, $b$, $c$, $d$ modulo $p$ as \begin{align*} N_G(P) &= \sum_{a,b,c,d} \...

4

I think generalizing a conjecture we know sol little about is a risky business, but let me try to say something non-vacuous. First of all, I'm assuming that $E$ has good ordinary reduction (otherwise, it has a $\mu$-invariant, but of a different kind and I don't think that this is what you have in mind - tell me if I'm wrong). In that situation, what do we ...

4

They are related because they come from the same element for $X_1(N)$. Suppose $E$ is the one of the two with the smaller degree of the modular parametrisattion $X_1(N)\to E$ of minimal degree. Then the isogeny $E\to E'$ sends the zeta elements from $E$ to the corresponding zeta elements for $E'$. This is by definition essentially. Conjecturally $E\to E'$ ...

4

A flippant response is that people had the idea of using perfectoid theory in Iwasawa theory long before perfectoid theory even existed. What I'm referring to here is the work of Fontaine--Wintenberger, who studied the "field of norms" of a tower of p-adic fields, or the "tilt" as youngsters like you would call it, way back in the 1970's. Scholze's tilting ...

4

I do not think there is a reference for this theory, because as far as I know no such theory exists. I have spent a substantial portion of my career studying the arithmetic of the special values of the $GL_2 \times GL_2$ Rankin--Selberg L-function, and I am not aware of a theory of modular symbols in this setting.

4

There is no reason why you shouldn't consider completed etale cohomology, instead of completed singular cohomology. If you look at Emerton's 2006 Inventiones paper which started the whole theory, he allows etale cohomology as well, in order to get a Galois action on completed cohomology; and there is a huge industry of studying completed cohomology spaces as ...

4

I haven't thought about base changes, but the original problem for $\alpha=1$ (so $\sigma$ is the identity map and $R=\mathbb{Z}_p[[t]][F]$ is just a polynomial ring over $\mathbb{Z}_p[[t]]$, and classifying $R$-modules that are finitely generated and free over $\mathbb{Z}_p[[t]]$ up to isomorphism is equivalent to classifying square matrices over $\mathbb{Z}... 3 I don't think we know how to prove this directly. Indeed, recent works by Wake and Wake–Erickson show that this cyclicity is equivalent to a conjectured improvement of Mazur–Wiles' result to the effect that a suitable localization of the Hecke algebra$\mathfrak{H}_\mathfrak{p}$is Gorenstein. The conjuction of the two papers shows how Greenberg's ... 3 Better not to prove it, as it is wrong. Writing$M[p] = \lbrace m \in M: pm = 0 \rbrace$for the exact$p$-torsion of a module$M$, I understand your question as follows: If the exact sequence $$0 \rightarrow A[p] \rightarrow A \rightarrow A/A[p] \rightarrow 0$$ remains exact modulo$p$(which is not always the case: it is equivalent to$A[p] \cap pA = 0$), ... 3 I'm assuming$\Lambda=\mathbf{Z}_p[[\mathbf{Z}_p]]\cong\mathbf{Z}_p[[T]]$(with a topological generator$\gamma$of$\Gamma=\mathbf{Z}_p$going to$1+T$). Let$Z$be the$\mathbf{Z}_p$-torsion submodule of the$\Lambda$-submodule$X^\prime=\bigcup_{n\geq 0}X^{\Gamma^{p^n}}$. Since$X^\prime$is finitely generated over$\Lambda$,$X^\prime=X^{\Gamma^{p^n}}$... 3 The article Sur les ideaux dont l'image par l'application d'Artin dans une$\mathbb{Z}_p$-extension est triviale of Michel Emsalem provides a satisfactory answer to the general question of how many places can split in a$\mathbb{Z}_p\$-extension. As such, I will mark this question as answered.

2

Regarding 1), Robert Pollack's tables - http://math.bu.edu/people/rpollack/Data/data.html - will be probably useful.

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