All Questions
24 questions
86
votes
8
answers
13k
views
What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?
Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
- 41.4k
23
votes
2
answers
2k
views
Even Galois representations "mod p"
Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...
- 17.6k
18
votes
1
answer
564
views
To what extent are modular parametrizations expected to generalize?
By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...
- 281
18
votes
0
answers
1k
views
Automorphic forms and coherent cohomology
Why is it (and what does it mean) that automorphic forms do not contribute in the coherent cohomology of Siegel modular varieties parametrizing abelian varieties of dimension $d>2$ (see section 7 ...
- 3,309
13
votes
2
answers
781
views
Elliptic curves and supercuspidal representations of conductor $p^2$
Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...
- 20.8k
13
votes
1
answer
1k
views
Reference for: CM Hilbert Modular forms arise from Hecke characters
For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
- 659
13
votes
0
answers
366
views
Lifting automorphic Galois representations to arithmetic fundamental groups and their quotients
Suppose $V$ is an algebraic variety over a number field $K$. The absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the étale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$ is ...
user127738
13
votes
0
answers
2k
views
Why doesn't functoriality immediately imply the modularity theorem?
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...
- 7,062
11
votes
1
answer
646
views
Modularity of higher genus curves
The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field.
What ...
- 1,267
11
votes
1
answer
660
views
Finiteness or infiniteness for Galois representations with unusual Hodge numbers
Say a representation $\operatorname{Gal}(\mathbb Q) \to GL_n(\overline{\mathbb Q}_\ell)$ has big monodromy if the Zariski closure of the image of $\operatorname{Gal}(\mathbb Q) $ contains $SO_n$ or $...
- 148k
10
votes
2
answers
1k
views
Automorphic forms and Galois representations over imaginary quadratic fields: generalizing Taylor's theorem?
Let $K/\mathbf{Q}$ be an imaginary quadratic field, with $\sigma \in \mathrm{Gal}(K/\mathbf{Q})$ a generator. Suppose $\pi$ is a cuspidal automorphic representation of $GL_2 / K$ with central ...
- 13.1k
8
votes
0
answers
335
views
Irreducibility of Galois representations attached to unitary groups
If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...
- 26k
7
votes
0
answers
256
views
Galois representations associated to the modular tower and automorphy
Consider $\mathcal{M} = \{M_{g,n}, \mu_{g,n}^{g’,n’}\}$, the system of moduli spaces of $n$-pointed smooth algebraic curves along with the basic maps amongst them coming from identifying and ...
user122285
6
votes
1
answer
612
views
Converse to Modularity II: Maass cusp forms
(This comes from this other question. You can find more details there)
The following bijection is now a theorem:
Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
newforms
note: ...
- 17.6k
6
votes
1
answer
499
views
Galois representation and weight one Hilbert modular form
Let $f$ be a primitive weight one Hilbert modular form for the totally real field $F$ (the weight of $f$ is parallel). Assume that $f$ is $N$-new, $p$-stable and cuspidal. Let $\rho:G_F \rightarrow \...
- 1,066
6
votes
0
answers
248
views
Galois action on functions on an adelic coset space
For a reductive group $G$ over a number field $K$, an automorphic representation for $(G, K)$ is an irreducible admissible constituent of the right regular representation of $G(\mathbb{A}_K)$ in the ...
user122285
5
votes
1
answer
289
views
Irrelevant parabolics and inner forms of GSp(4)
In Ralf Schmidt's appendix to "Jacquet-Langlands-Shimizu correspondence for theta lifts to $\mathrm{GSp}(2)$ and its inner forms" by Narita and Okazaki , he computes the representations of $\mathrm{...
- 2,429
3
votes
0
answers
137
views
Field of definition of automorphic Galois representation
Let $\pi$ be a regular, cuspidal, algebraic automorphic representation of $GL_n(\mathbb{A}_K)$ for a totally real field $K$. Then for every embedding $\lambda$ of $E=\mathbb{Q}(\pi)$ in $\overline{\...
- 190
3
votes
0
answers
740
views
The operator \boxtimes and \boxplus in automorphic representations
Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$.
Now, for each $v$, let $\pi_{1, v}\boxtimes \pi_{...
- 31
2
votes
1
answer
491
views
Newform and Galois representation (Shimura-Deligne Reciprocity Law)
Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon {\mathrm{...
- 101
2
votes
2
answers
1k
views
Decomposition of Artin L functions
The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of $Gal(...
- 11.2k
2
votes
1
answer
247
views
Galois representations attached to a cusp form for different primes
If I have a cusp form $f$, I can consider the associated Galois representation $\rho_l(f)$ for any prime $l$. For two distinct primes $p$ and $q$, what is the relationship between $\rho_p(f)$ and $\...
- 21
2
votes
0
answers
94
views
Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The ...
- 6,622
0
votes
0
answers
116
views
Gauss lemma for a complete Noetherian domain
Suppose that $R$ is a Noetherian complete domain over a field $K$.
Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...
- 563