# Questions tagged [hida-theory]

The hida-theory tag has no usage guidance.

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### Existence of newforms which are non-ordinary at a given prime

Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a_p(f)$ is a $p$-adic unit (...

**4**

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**1**answer

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### Example of two p-Ordinary Elliptic Curves congruent to each other

I am looking for an example of a prime, p, for which there exists two $p$-ordinary rational elliptic curves $E$, $F$ for which, at every prime $l$ not dividing $N=p \operatorname{Cond}(E) \...

**6**

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254 views

### What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...

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147 views

### Kitagawa's p-adic modular symbols for different weights: a confusing observation

References are to K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms", Contemp. Math. 165, 1994.
Let $\mathcal O$ be the ring of integers in a finite extension of $\...

**2**

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210 views

### Control theory for Kitagawa's $\Lambda$-adic modular symbols

Let $p$ be a prime, $\Gamma=1+p\mathbb Z_p$ and $\Lambda=\mathbb Z_p[[\Gamma]]$ the Iwasawa algebra. Let $\kappa\colon\Gamma\rightarrow\mathbb Z_p^\times$ be the inclusion. For a character $\...

**2**

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**1**answer

212 views

### Families of ordinary Siegel Modular Forms

I'm looking for references to constructions and treatments of Hida Families/Eigenvarieties for ordinary Siegel modular forms (In particular: genus 2).
So far I've been reading Richard Taylor's thesis ...

**9**

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**1**answer

458 views

### Arithmetic Points are Dense on a Hida Family

I am reading the paper "Constancy of the Adjoint L-invariant" by H. Hida (http://www.math.ucla.edu/~hida/ConstP.pdf).
Correct me if I'm wrong, but I've read/heard that the arithmetic points $p \in ...

**9**

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**1**answer

582 views

### Definition of p-adic modular forms

I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...

**5**

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**2**answers

268 views

### Examples of component crossing between families of modular forms

Is there a reference that contains explicit examples of component crossing of Hida families at height one primes? The paper of Emerton, Pollack, and Weston addresses component crossing obtained ...

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### Intersection of ordinary subspaces at different primes

Choose two distinct primes $\ell$ and $\ell'$, and embeddings $\iota_\ell : \overline{\mathbb Q} \to \overline{\mathbb Q_\ell}$, $\iota_{\ell'} : \overline{\mathbb Q} \to \overline{\mathbb Q_{\ell'}}$....

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### Atkin--Lehner operators in Hida theory

Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner ...

**10**

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**1**answer

465 views

### Example of a non-smooth irreducible component of the generic fibre of a Hida family?

Is there a known example of a non-smooth irreducible component of the rigid generic fibre of a Hida family?
Let me explain some of the context around this question (but I'm not going to explain Hida ...

**3**

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**2**answers

487 views

### Interpolation of periods for a Hida family of modular forms

Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit
$$ \...

**7**

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**2**answers

866 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 ...

**14**

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**2**answers

853 views

### Examples of q-expansions in a Hida family

Let $p$ be a prime number and $N$ a positive integer not divisible by $p$.
For some easy choices of $p$ and $N$, can anybody provide me with explicit examples of collections $$\{f_k,\quad 2\leq k \...

**9**

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**2**answers

633 views

### Non-classical specializations of Hida families

Let ${\mathbb T}$ denote the ordinary $\Lambda$-adic Hecke algebra of say tame level $N$. If I specialize ${\mathbb T}$ to a classical weight $k \geq 2$, then it is proven by Hida that the result is ...

**7**

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**3**answers

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### Free subquotient of Galois representations coming from Hida theory

Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf{...