Questions tagged [siegel-modular-forms]
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27
questions
8
votes
0answers
260 views
Boundary of Siegel modular variety
The moduli space of curves has a compactification whose boundary can be understood as the product of moduli spaces of curves of lower genus. Therefore (perhaps naively) one might hope that there ...
2
votes
0answers
142 views
Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields
Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...
5
votes
0answers
175 views
Diophantine applications of Paramodularity
I’ve asked this question to quite a few people in person and so far haven’t seen a good answer...but I believe one should exist, so here goes!
Ok, we all know how to (roughly) prove Fermat’s Last ...
6
votes
0answers
320 views
Integral models of perfectoid modular curves
Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–...
6
votes
1answer
416 views
Behavior of a modular form in the lower strip
Let $f$ be an (elliptic) modular form of weight $k>0$, and consider the vertical strip $S_m=\{x+iy\in\mathbb{C}:|x|\le 1/2, y>m$}. For every $m\ll 1$, the fundamental domain for $SL_2(Z)$ is ...
5
votes
1answer
247 views
Hecke operators for hermitian modular forms of general level
It has bug me for a while that I don't have a good understanding of the theory of Hecke operators. For elliptic modular forms, it was explained in Koblitz's book that they arose from viewing the ...
3
votes
1answer
194 views
Non holomorphic Siegel-Poincare series growth
Let $H_g$ be the Siegel upper-half plance of genus $g$, and $\Gamma=Sp_{2g}(\mathbb{Z})$ the full modular group of genus $g$. As usual, we can define non-holomorphic Eisenstein and Poincare series as
$...
2
votes
0answers
85 views
Growth of a modified Zeta function appearing in the non-holomorphic Siegel Eisenstein series
In a paper (Eisenstein series for Siegel modular groups, https://link.springer.com/content/pdf/10.1007/BF01459520.pdf) Mizumoto obtains an explicit Fourier expansion for the non-holomorphic Siegel ...
10
votes
1answer
349 views
Special values of adjoint $L$-functions of automorphic representations of $\mathrm{GSp}(4)$ as Petersson norms
Here I consider cuspidal automorphic representations $\pi$ over the similitude group $\mathrm{GSp}(4,\mathbb{A}_\mathbb{Q})$. Let $f$ be a non-zero vector in the representation $\pi$. I want to know ...
3
votes
0answers
182 views
Hecke operator acting on Siegel modular forms
Let $F,G$ be Hecke eigenforms of weight $k$ and genus $2$. For any Hecke operator $T$ (either $T_q$ or $T_{q^2}$) let $\lambda_T(\star)$ be the correspondent eigenvalue.
Assume that there exists a ...
7
votes
1answer
236 views
Fourier expansion of the Saito-Kurokawa lift
As is well known, the Saito-Kurokawa lifts maps (classical) cusp forms $f$ to Siegel (genus 2) cusp forms $SK(f)$.
Is there an explicit formula for the Fourier expansion of a Saito-Kurokawa lift?
...
2
votes
1answer
179 views
Semistability of local Siegel Galois rep:
When are the $l$-local $p$-adic Galois representations of Siegel modular forms semistable? By this I mean $\rho_{f}: G_{\mathbb{Q}}\to \operatorname{GSpin}_{2n+1}(\overline{\mathbb{Q}}_p)$ restricted ...
6
votes
1answer
221 views
On the local automorphic components of classical Siegel modular forms
I am looking for a dictionary that relates the level of a classical genus 2 Siegel modular form and the local components of the corresponding automorphic representation of $Gsp_4(\mathbb{A}_{\mathbb{Q}...
2
votes
0answers
401 views
Wedge product of entries of a matrix & Volume form of the Siegel metric
Let $A=(a_{ij})$ be an $n\times n$ square matrix, and $\omega(A)=\bigwedge\limits_{i,j=1}^na_{ij}$ be the wedge product of its entries. Then, if $B=UA=(b_{ij})$ for some square matrix $U$, I think one ...
1
vote
0answers
125 views
Reference: Heat Kernel for Siegel Upper Half plane
Is there a ready reference for explicit computation of the heat kernel for Siegel upper half space $\mathbb{H}_n=\{Z=X+iY\in \mathrm{Mat}_n(\mathbb{C}) \vert Y>0\} $? I could find it for general ...
2
votes
0answers
101 views
Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane
Let $A,B\in M_n(\mathbb{R})$ and $U=A+iB$ unitary. $R=diag(r_1,r_2,…,r_n)$ is a diagonal matrix with $r_i>0, \forall i $. I need to calculate $\det(Ae^{-R}A^T+Be^{R}B^T)$. This matrix $Ae^{-R}A^T+...
2
votes
1answer
263 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 ...
6
votes
1answer
328 views
A Siegel modular form related to the product of two eta functions
I am looking for a Siegel modular form of genus $2$ (living on the Siegel modular 3-fold $A_2=\mathrm{Sp}(4,\mathbb{Z})\backslash \mathfrak H_2$) which becomes "roughly" the product of two eta ...
2
votes
2answers
243 views
Connection between the two definitions of Siegel Upper Half Space
It seems, there are two definitions of the Siegel upper half space.
1) One used by, say, Krantz in "Explorations in Harmonic Analysis; Page 252, or, by So Chin Chew and Mei Chi Shaw in "Partial ...
8
votes
0answers
283 views
P-depletion of Siegel modular forms
Let $F$ be a cuspidal Siegel modular form of genus 2 (of parallel weight $(k, k)$, and level some congruence subgroup $\Gamma \subseteq Sp_4(\mathbf{Z})$ of level $N$).
Then $F$ has a series ...
5
votes
1answer
341 views
Index of congruence modular subgroup of level (1,d)
Let $D = \text{diag}(1,d)\in M_{2}(\mathbb{Z})$ be a $2\times 2$ matrix, where $d$ is an odd integer. We define the subgroup $\Gamma_D\subset M_{4}(\mathbb{Z})$ as:
$$\Gamma_D := \left\lbrace R\in M_{...
4
votes
0answers
134 views
Generators of the symplectic subgroup $\Gamma^g(1,2)$
Let $\mathbb{A}^{m\times n}$ denote the set of all $m \times n$ matrices with entries in the set $\mathbb{A}$. For a matrix $M$ we let ${^tM}$ denote its transpose, and $M^{-1}$ its inverse, if it is ...
10
votes
0answers
400 views
P-stabilization of Siegel modular forms
Here's a well-known lemma about modular curves:
Let $\pi_1, \pi_2$ be the two degeneracy maps $Y_1(Np) \to Y_1(N)$, for $p \nmid N$, corresponding to $z \mapsto z$ and $z \mapsto pz$. Then as ...
4
votes
0answers
383 views
Motivic interpretation of genus 2 Siegel forms induced by lifts of Maass and Skoruppa
Background: There are several known lifts from integral weight modular forms to Siegel forms of genus 2, among them the Saito-Kurokawa lift. Another lift construction that is important for ...
4
votes
4answers
1k views
Siegel modular forms as sections of line bundles over the period domain
The transformation formula for a Siegel modular form can be interpreted as the statement that the modular form is a holomorphic section of a line bundle over the period domain (the quotient of the ...
12
votes
1answer
1k views
Analogue of Shimura curves in the symplectic case?
My understanding is this: one can attach 2-d Galois representations to classical modular eigenforms because one can look in the etale cohomology of modular curves. For Hilbert modular forms the naive ...
18
votes
0answers
873 views
Computation of low weight Siegel modular forms
We have these huge tables of elliptic curves, which were generated by computing modular forms of weight $2$ and level $\Gamma_0(N)$ as N increased.
For abelian surfaces over $\mathbb{Q}$ we have very ...
