# Questions tagged [siegel-modular-forms]

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27
questions

**8**

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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**

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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

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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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

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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

**1**answer

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**

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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

**1**answer

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

**1**answer

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

**1**answer

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

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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

**0**answers

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

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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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

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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**

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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**

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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

**4**answers

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

**1**answer

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**

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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 ...