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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 ...
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0answers
80 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+...
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1answer
145 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
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1answer
233 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 ...
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2answers
104 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 ...
6
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0answers
190 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
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1answer
241 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_{...
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89 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 ...
7
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0answers
208 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 ...
3
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0answers
342 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 ...
3
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3answers
854 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
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1answer
888 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 ...
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0answers
779 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 ...