# Tagged Questions

**1**

vote

**1**answer

174 views

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

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$.
Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...

**3**

votes

**2**answers

140 views

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

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...

**6**

votes

**1**answer

666 views

### Main conjecture for elliptic curves

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...

**4**

votes

**2**answers

292 views

### Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$

**3**

votes

**1**answer

262 views

### Splitting as $\mathbb{F}_p[[X]]$-modules

Let $A$ be a finitely generated torsion $\mathbb{Z}_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB ...

**3**

votes

**1**answer

362 views

### $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2

Consider the elliptic curves -
$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $
$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $
with both good ...

**4**

votes

**2**answers

213 views

### $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 1

1) What are the examples of elliptic curves over $\mathbb{Q}$ with good reduction and $\mu$-invariant $\geq 2$ at $p = 3$ and how to find them $?$
2) Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ ...

**2**

votes

**1**answer

364 views

### $\lambda$-invariant is constant for isogenous elliptic curves

How to prove that the $\lambda$-invariant is constant for isogenous elliptic curves $?$

**7**

votes

**2**answers

514 views

### Periods for 2-variable p-adic L-functions

Hi all,
I am sorry to ask a stupid question but I am really confused right now. Kitagawa-Mazur constructed a $2$-variable p-adic L-function attached to Hida families of modular forms. For their ...

**3**

votes

**1**answer

534 views

### Applications of Iwasawa Theory

Iwasawa theory gives a formula for the power of $p$ dividing the class group of the $\mathbb{Q}(\zeta_{p^n})$ (where $\zeta_{p^n}$ is a primitive root of unity of exact order $p^n$) for sufficiently ...

**1**

vote

**1**answer

320 views

### main conjecture of Iwasawa theory implies Herbrand-Ribet

I need a reference/proof for "main conjecture of Iwasawa theory => refinement of Herbrand-Ribet ($v_p(B_{p-i}) = v_p(|A_i|)$, where $A_i$ denotes the $i$-th eigenspace of the Galois group acting on ...

**10**

votes

**2**answers

641 views

### Geometric interpretation of Hida isomorphism

[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely.
As the title says, I would like to understand ...

**3**

votes

**1**answer

525 views

### Gouvea-Mazur conjecture

I was reading about the conjecture made by Gouvea and Mazur in their paper "Families of modular eigenforms" which says that if $k_1 \equiv k_2 \pmod {p^{n}(p-1)}$ for some integer $n\geq \alpha$. then ...

**15**

votes

**1**answer

1k views

### Two-variable p-adic L-functions of elliptic curves

Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$.
If $E / \mathbb{Q}$ is an ...

**1**

vote

**2**answers

649 views

### Functional equations relating to p-adic L-functions

Let f be a modular form of weight k for $\Gamma_0(N)$. Let us assume that $p\not\vert$N. Then we can construct 2 p-adic L-functions corresponding to the 2 roots $\alpha$ and $\beta$ of the equation ...

**13**

votes

**3**answers

938 views

### rational points on algebraic curves over $Q^{ab}$

Motivation:
Let $Q_{\infty,p}$ be the field obtained by adjoining to $Q$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyclotomic extension ...

**15**

votes

**3**answers

1k views

### Non-vanishing of p-adic L-functions

In Non-vanishing of L-series of modular forms (easy case?) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is ...

**8**

votes

**1**answer

568 views

### Iwasawa mu-invariant for abelian extensions of quadratic number fields

Let K be a number field and $p$ an odd prime. Let $\mu$ be the Iwasawa $\mu$-invariant of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $K$. If $K$ is abelian over $\mathbb{Q}$ then it ...

**11**

votes

**1**answer

888 views

### Fontaine's classification of p-divisible groups

Let k be a finite extension of $\mathbb{F}_p$, and W its ring of Witt vectors. Write W[F,V] for the Dieudonn\'e ring.
Let G be a connected p-divisible group which is finite-dimensional over k, and ...

**22**

votes

**1**answer

928 views

### Does the everywhere unramified extension of Q(mu_37) of degree 37 grow into a Z_37-extension?

Let $p=37$. Since $p$ divides the numerator of $B_{32}$, by Ribet's proof of the converse of Herbrand's theorem, we know that the class group of ${\bf Q}(\mu_p)$ has size divisible by $p$. More ...

**10**

votes

**2**answers

1k views

### Are Kato's zeta elements integral?

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 ...

**14**

votes

**2**answers

1k views

### P-adic L functions

My question is how should one think of p-adic L functions? I know they have been constructed classically by interpolating values of complex L-functions. Recently I have seen people think about them in ...

**4**

votes

**1**answer

469 views

### Characteristic Complexes in Iwasawa theory

For all that follows, $p$ is a fixed odd prime. In the formulation of the Noncommutative Main Conjecture of Iwasawa theory one uses étale cohomology to define an algebraic object analogous to Iwasawas ...

**12**

votes

**1**answer

547 views

### Capitulation in cyclotomic extensions

Let $p$ be an irregular prime, which means that $p$ divides some Bernoulli number: $p \mid B_k$ (for some even $k\in[2,p-3]$). This implies that the class number of the field $K$ of $p$-th roots of ...

**9**

votes

**1**answer

1k views

### A question about Iwasawa Theory

I am just reading about Iwasawa theory about Coates and Sujatha's book on Iwasawa Theory. I was wondering that since Iwasawa thought about the whole theory from the analogy of curves over finite ...

**12**

votes

**1**answer

715 views

### P-adic L-functions of nonabelian twists of elliptic curves

Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...

**7**

votes

**2**answers

1k views

### non-commutative iwasawa theory

In commutative Iwasawa theory, the main conjecture states that the p-adic L-function generates the characteristic ideal of an algebraic object. Non-commutative Iwasawa theory seems to mimik this - ...

**7**

votes

**0**answers

513 views

### Existence of multi-variable p-adic L-functions

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields?
More precisely: if K is a number field, and $K_{\infty} / K$ is an infinite Galois extension, ...

**10**

votes

**5**answers

1k views

### When does a p-adic function have a Mahler expansion?

Let $f: \mathbb{Z}_p \rightarrow \mathbb{C}_p$ be any continuous function. Then Mahler showed there are coefficients $a_n \in \mathbb{C}_p$ with
$$
f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}.
$$
...

**14**

votes

**4**answers

2k views

### Special values of $p$-adic $L$-functions.

This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.
My understanding is that nowadays there are conjectures which essentially ...

**13**

votes

**1**answer

863 views

### von Staudt-Clausen over a totally real field

Before I ask the question, I need to recall what Bernoulli numbers
$(B_k)_{k\in\mathbb{N}}$ are, and what von Staudt and Clausen discovered about
them in 1840. The numbers $B_k\in\mathbb{Q}$ are the ...

**17**

votes

**4**answers

1k views

### Does p-adic $L$- function determine the $L$ function

Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic ...

**20**

votes

**7**answers

3k views

### Model theoretic applications to algebra and number theory(Iwasawa Theory)

One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...

**22**

votes

**6**answers

4k views

### How do we study Iwasawa theory?

What papers should we read to start? What basic knowledge do we need to understand the question? What is this area really about? And what are people researching on it? Thank you!