# Questions tagged [hida-theory]

The hida-theory tag has no usage guidance.

24
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### When do Fourier coefficients vanish in Hida families?

Suppose you have a Hida family with $q$-expansion $F = \sum_{n=1}^{\infty} a_n(T) q^n$, where the coefficients $a_n(T)$ are power series in $\mathbb{Z}_p [[T]]$. Assume that $F$ is a cuspidal ...

1
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1
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### Reference for auto-duality of nearly ordinary deformations associated to Hida families

suppose we have a $p$-stabilized newform $f$ of classical level $\Gamma_0(p)$; then there is a unique ordinary deformation into a Hida family $\mathbf{f}$ defined over a finite flat extension $R/\...

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### Motivation for $p$-stabilization in Hida theory

I'm currently reading Hida's paper "A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms". The setup is the following: let $f$ be a weight $2$ newform ...

3
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0
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### Pilloni's cohomological corrispondence factorization

I'm trying to understand the proof of Lemma 7.1.1 at page 39 of Pilloni's paper on Higher Hida and Coleman Theory for $GSp_4$. In particular, what is not clear to me is the diagram relating Serre-Tate'...

3
votes

1
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### Existence of congruences between modular forms / elliptic curves

I'd like to ask two questions about congruences: one about modular forms and one about elliptic curves.
Suppose we are given a cusp form $f$ of weight $2$ and level $\Gamma_0(N)$. Given a good ...

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### What is "the quotient of the universal ordinary Hecke algebra corresponding to an ordinary $\Lambda$-adic form"?

Let $\Lambda := \Bbb Z_p[[T]]$ be the usual Iwasawa algebra. In Jha and Sujatha - On the Hida deformations of fine Selmer groups on page 181, the authors refer to the quotient $\Bbb H^{\text{ord}}_{\...

12
votes

1
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484
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### Eigenvarieties and functoriality

In Langlands' review of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces ...

8
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446
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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 (...

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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) \...

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### 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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168
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### 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 $\...

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252
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### 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
votes

1
answer

314
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### 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 ...

11
votes

2
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### Arithmetic points are dense on a Hida family

$\DeclareMathOperator\Spec{Spec}$I am reading the paper "Constancy of the adjoint L-invariant" by H. Hida (J. Number Th., 2011, DOI link).
Correct me if I'm wrong, but I've read/heard that ...

9
votes

1
answer

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### 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 ...

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votes

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answers

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### 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 ...

2
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### Intersection of ordinary subspaces at different primes

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

11
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1
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759
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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

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### 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
votes

2
answers

602
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### 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
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### 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 ...

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votes

2
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### 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 \...

10
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2
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### 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 ...

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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{...