# Tag Info

Accepted

### Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?

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 $\frac{1}{ 1- |\mathfrak a|^{-1} }$ ...
• 116k
Accepted

• 31.5k
Accepted

### Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny

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.
• 31.5k
Accepted

### Refinement of (classical) Iwasawa main conjecture

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 ...

### Classification of cyclotomic fields with class number 1

The complete list of $n$ for which ${\bf Q}(\zeta_n)$ has unique factorization is 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90. Source; ...
Accepted

### Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension

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"
• 6,874
Accepted

### Some questions related to Iwasawa invariants of elliptic curves

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 ...
• 9,500

### Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny

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 ...
• 6,874
Accepted

### Elliptic curves and $GL(2)$ Iwasawa theory

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 $P$...
• 6,874

### Herbrand-Ribet and Mazur-Wiles for function fields

(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 ...
• 31.5k

### Kato's Euler System for Isogenous Elliptic Curves

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 ...
• 6,874
Accepted

### Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?

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 (...
• 9,500

### Iwasawa theory and perfectoid spaces

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--...
• 31.5k
Accepted

### Modular symbols associated to Rankin Selberg convolutions and the symmetric square

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 ...
• 31.5k

### Completed cohomology and variants

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 ...
• 31.5k

### Classification of finitely generated modules over non-commutative rings

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 ...
• 29.8k

### Value of the quadratic Gauss sum viewed in $\mathbb{C}_p$

$\DeclareMathOperator\sgn{sgn}$Just as in the complex case, we have that $G(\sgn, \chi)$ lies in $\mu_4(\mathbb C_p)\sqrt q$. We have that $G(\sgn, \chi)$ equals $\sum_{c \in \mathbb F_q} \chi(c^2)$, ...
• 7,923

### Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

Much more than Iwasawa's original theorem is known by now. First of all, the $p$-primary part of the class group stays trivial in the field $K_n=\mathbb{Q}(\mu_{p^{n+1}})$ unless it is already non-...
• 6,874
Accepted

You are asking about substitution of one power series into another. This can be interpreted in two ways, which ultimately amount to the same thing (like two different ways of thinking about anything) ...
• 41.8k
Accepted

### State of the art on the main conjecture for supersingular elliptic curves/modular forms

Theorem (Xin Wan): If $E$ is an elliptic curve of square-free conductor $N$, and $p \ge 3$ is a prime such that $p \nmid N$ and $a_p(E) = 0$, then Kobayashi's $\pm$ Iwasawa main conjectures are true ...
• 31.5k
### Completely split primes in non-anticyclotomic $\mathbb{Z}_p$-extensions
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 ...