Questions tagged [hida-theory]

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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 ...
Adithya Chakravarthy's user avatar
1 vote
1 answer
81 views

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/\...
xir's user avatar
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2 votes
0 answers
219 views

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 ...
Adithya Chakravarthy's user avatar
3 votes
0 answers
198 views

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'...
Fra's user avatar
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3 votes
1 answer
227 views

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 ...
Adithya Chakravarthy's user avatar
4 votes
0 answers
152 views

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}}_{\...
Edward Evans's user avatar
12 votes
1 answer
484 views

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 ...
user avatar
8 votes
1 answer
446 views

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 (...
François Brunault's user avatar
4 votes
1 answer
197 views

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) \...
Eins Null's user avatar
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6 votes
0 answers
415 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 ...
Will Dukeminier's user avatar
1 vote
0 answers
168 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 $\...
Michael Fütterer's user avatar
2 votes
0 answers
252 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 $\...
Michael Fütterer's user avatar
2 votes
1 answer
314 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 ...
Eins Null's user avatar
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11 votes
2 answers
667 views

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 ...
Eins Null's user avatar
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9 votes
1 answer
844 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 ...
Bear's user avatar
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5 votes
2 answers
363 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 ...
jkramerm's user avatar
  • 510
2 votes
0 answers
83 views

Intersection of ordinary subspaces at different primes

Choose two distinct primes $\ell$ and $\ell'$, and embeddings $\iota_\ell : \overline{\mathbb Q} \to \overline{\mathbb Q_\ell}$, $\iota_{\ell'} : \overline{\mathbb Q} \to \overline{\mathbb Q_{\ell'}}$....
Bruno Joyal's user avatar
  • 3,860
11 votes
1 answer
759 views

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 ...
David Loeffler's user avatar
10 votes
1 answer
528 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 ...
Kevin Buzzard's user avatar
3 votes
2 answers
602 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 $$ \...
David Loeffler's user avatar
7 votes
2 answers
1k 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 ...
Arijit's user avatar
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17 votes
2 answers
1k 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 \...
Victor Rotger's user avatar
10 votes
2 answers
785 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 ...
H P 's user avatar
  • 103
7 votes
3 answers
2k views

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{...
Olivier's user avatar
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