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Register a new userComputing genus 2 Hilbert-Siegel modular forms over $Q(sqrt{5})$ via the Jacquet-Langlands correspondence
تحويل نماذج هيلبرت-سايغل الجنس 2 في $Q(sqrt{5})$ عبر التطابق جاكيه-لانجلاندس
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في هذه الورقة، نقدم خوارزمية لحساب نظم الأيجنستيم الخاصة بالشكل الحجري الحلبيرت-سايجل لحقول المربعة الريالية الذاتية الثنائية من عدد الفئة الضيقة واحد. نعطي بعض الأمثلة البيانية باستخدام الحقل الثنائي $Q(sqrt{5})$. في هذه الأمثلة، نحدد الشكل الحجري الحلبيرت-سايجل الأيجنستيم الذي يمكن أن يكون رفعا من الشكل الحجري الأيجنستيم.
In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $Q(sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms.
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We study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications to the Langlands programme: Using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet-Langlands correspondence between $operatorname{GSp}_4$ and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $operatorname{GSp}_4$.
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke modules.
This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules by certain $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules. Here we let $F=mathbb{Q}_p$ (and $pgeq 5$) and construct a morphism from $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters to $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuils semisimple mod $p$ Langlands correspondence for $mathbf{GL_2}(mathbb{Q}_p)$ and of Pav{s}k={u}nas parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $mathbf{GL_2}(mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-representations of the Galois group ${rm Gal}(overline{mathbb{Q}}_p/ mathbb{Q}_p)$.
We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the etale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures.
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