All Questions
Tagged with automorphic-forms arithmetic-geometry
24 questions
2
votes
1
answer
164
views
What is the sum operation on torsors induced by Weil uniformization?
Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...
- 1,969
11
votes
1
answer
575
views
Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
- 639
3
votes
0
answers
202
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'...
- 91
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
4
votes
0
answers
410
views
What is a Shimura variety?
It so look like that Shimura variety is a modular space that contains classes of zeros of modular functions that have certain symmetry that have structure of Hodge type.
Is this true?
9
votes
1
answer
787
views
The cohomology of modular curves as a module over the Galois group
Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \...
- 7,746
11
votes
0
answers
359
views
Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$
Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...
- 7,746
5
votes
1
answer
435
views
Fourier coefficients of Siegel Eisenstein series
I am looking for reference about Fourier coefficients of Eisenstein series. Currently I am mainly interested Eisenstein series given by Siegel parabolic subgroup of $SP_{2n}$ and $U(n,n)$. Let's ...
- 409
1
vote
0
answers
90
views
Is there an analogue of theta cycles for more general mod p automorphic forms?
The theory of $\theta$-cycles (due to Tate, I think) and filtrations is to me a very beautiful and powerful tool in proving many statements about mod $p$ modular forms in more explicit and elementary ...
- 1,024
4
votes
0
answers
206
views
Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?
In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k}...
- 6,622
26
votes
1
answer
959
views
What automorphic forms are expected to occur in the zeta function of moduli space of curves?
Assume $g \geq 1$ and $n \geq 0$, the moduli stack ${\mathcal {M}}_{g,n}$ classifies families of smooth projective curves of genus $g$ with $n$ marked points , together with their isomorphisms. It has ...
- 6,622
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
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
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
4
votes
2
answers
400
views
Potential automorphy of abelian varieties
Let $A$ be an abelian variety over $\mathbb Q$. One could ask
(1) is there a finite extension $K$ of $\mathbb Q$ such that the L-function $L(A/K,s)$ is the L-function of an automorphic form?
or
...
- 528
12
votes
0
answers
285
views
Modularity of endomorphism algebras
This question is about comparing Hecke algebras and endomorphism algebras.
Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...
- 20.8k
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
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
12
votes
2
answers
1k
views
Modularity of higher dimensional abelian varieties
In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, one can give a proof using ...
- 7,062
5
votes
1
answer
630
views
Special value of $L$-function
Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...
- 471
3
votes
1
answer
667
views
What is an automorphic representation of CM type ?
In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a ...
- 809
5
votes
3
answers
973
views
The historical development of automorphic geometry
Background:
Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
- 1,704
45
votes
2
answers
3k
views
Langlands in dimension 2: the Yoshida conjecture
Background:
One prominent part of the Langlands program is the conjecture that
all motives are automorphic.
It is of interest to consider special cases that are more precise, if less
sweeping. ...
- 1,704
1
vote
2
answers
1k
views
Arithmetic geometry from a bird's-eye view
Is ist true that Arithmetic Geometry can roughly be separated into two areas:
1) Showing that motivic $L$-functions are automorphic.
2) Calculating special values of these $L$-functions.
user19475