All Questions
15 questions
5
votes
1
answer
891
views
Can Taniyama-Shimura conjecture be generalized to curves of higher genus (within Langlands framework)?
The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then there is an integer N ≥ 1 and a weight-two cusp form f of level N, normalized so that a1(f) = 1, such that ap(E)...
- 2,216
4
votes
0
answers
187
views
Understanding Shimura correspondence in context of Langlands functoriality
Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...
- 2,215
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
5
votes
0
answers
1k
views
Some questions about cuspidal representations and automorphic representations
My reference is Daniel Bump's book, Automorphic Forms and Representations. $G$ is a connected reductive group over a number field $k$ (in Bump's book he takes $G = \operatorname{GL}_n$). Let $K = K_{...
- 6,180
6
votes
2
answers
474
views
Symmetric powers of Ramanujan tau-function
Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.
For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$...
- 3,804
3
votes
0
answers
299
views
Equivalence of formulations of Ihara's lemma
I'm wondering about the relationship between two formulations of Ihara's lemma for $\text{GL}_2$ I've seen:
(1) the "concrete" version given in, for example, Darmon, Diamond, and Taylor, which says ...
- 2,044
7
votes
1
answer
811
views
Alternative way to prove the functional equation for Eisenstein series?
Let $E(z,s):=\pi^{-s}\Gamma (s) \sum_{(m,n)=1}\frac{y^s}{|mz+n|^{2s}}$ be the real-analytic Eisenstein series.
It satisfies the functional equation $E(z,s)=E(z,1-s)$ with two poles at $s=0,1$.
The ...
- 3,804
4
votes
1
answer
313
views
Is the twisted symmetric fifth power $L$-function holomorphic?
Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character.
Let us consider the $L-$ function
$$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times \...
- 3,804
9
votes
0
answers
435
views
Rankin-Selberg for Maass form GL(3)xGL(2)
Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character).
Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...
- 3,804
5
votes
1
answer
344
views
looking for reference on dihedral, tetrahedral, or octahedral forms
I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to $...
- 131
12
votes
2
answers
781
views
Langlands' original observation about Ramanujan conjecture
Obviously functoriality of arbitrary high symmetric power lifts of automorphic forms on GL(2) will lead to the Ramanujan conjecture. But I guess that is too strong for Ramanujan. I came across some ...
- 585
9
votes
1
answer
530
views
standard zero free region of automorphic L-function on GL(N)
Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.
What's the standard zero-free region for $L(s,\pi)$? any ...
- 3,804
0
votes
1
answer
852
views
Euler product of Asai L-function?
Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function
$$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$
where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...
- 3,804
3
votes
0
answers
222
views
Functoriality for triple product GL(2) x GL(2) x GL(2)
Let $f$, $g$ and $h$ be three general automorphic forms on $\operatorname{GL}(2)$.
Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on $\operatorname{GL}(8)$?
- 3,804
2
votes
0
answers
212
views
Conceptual reason behind Shimura lifts
Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G =SL_2(\mathbb{R}...
- 11.2k