CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE

2020 ◽  
pp. 1-22
Author(s):  
JIM BROWN ◽  
HUIXI LI
Keyword(s):  
Functional Equation ◽  
Modular Forms ◽  
Class Number ◽  
Automorphic Forms ◽  
Siegel Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Free Level ◽  
Totally Real

Abstract It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

scholarly journals Minimal weights of Hilbert modular forms in characteristic

2017 ◽  
Vol 153 (9) ◽  
pp. 1769-1778 ◽  
Author(s):  
Fred Diamond ◽  
Payman L Kassaei
Keyword(s):  
Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties ◽  
Totally Real Field

We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

scholarly journals GEOMETRIC WEIGHT-SHIFTING OPERATORS ON HILBERT MODULAR FORMS IN CHARACTERISTIC p

2021 ◽  
pp. 1-60
Author(s):  
Fred Diamond
Keyword(s):  
Modular Forms ◽  
Point Of View ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Prior Work ◽  
Characteristic P ◽  
Hilbert Modular ◽  
Totally Real ◽  
Weight Shifting ◽  
Geometric Weight

Abstract We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author’s prior work with Sasaki to the case where p is ramified in the totally real field. In particular, we use the partial Hasse invariants and Kodaira–Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren’s construction of partial $\Theta $ -operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore, we describe the kernels of partial $\Theta $ -operators in terms of images of geometrically constructed partial Frobenius operators. Finally, we apply our results to prove a partial positivity result for minimal weights of mod p Hilbert modular forms.

On adelic automorphic forms with respect to a quadratic extension

2000 ◽  
Vol 62 (1) ◽  
pp. 29-43 ◽  
Author(s):  
Ze-Li Dou
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Quadratic Extension ◽  
Algebraic Number Field ◽  
Hilbert Modular Forms ◽  
Arithmetic Properties ◽  
Theta Lift ◽  
Hilbert Modular ◽  
Algebraic Foundation ◽  
Totally Real

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

scholarly journals PARALLEL WEIGHT 2 POINTS ON HILBERT MODULAR EIGENVARIETIES AND THE PARITY CONJECTURE

2019 ◽  
Vol 7 ◽  
Author(s):  
CHRISTIAN JOHANSSON ◽  
JAMES NEWTON
Keyword(s):  
Modular Forms ◽  
Weight Space ◽  
Small Radius ◽  
Real Field ◽  
Central Character ◽  
Hilbert Modular Forms ◽  
Irreducible Components ◽  
Hilbert Modular ◽  
Parity Conjecture ◽  
Totally Real

Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension $1+[F:\mathbb{Q}]$ ) cuspidal Hilbert modular eigenvariety has the property that many (all, if $[F:\mathbb{Q}]$ is even) irreducible components contain a classical point with noncritical slopes and parallel weight $2$ (with some character at $p$ whose conductor can be explicitly bounded), or any other algebraic weight.

scholarly journals The Jacquet–Langlands correspondence for overconvergent Hilbert modular forms

2019 ◽  
Vol 15 (03) ◽  
pp. 479-504 ◽  
Author(s):  
Christopher Birkbeck
Keyword(s):  
Modular Forms ◽  
Quaternion Algebra ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Langlands Correspondence ◽  
Hilbert Modular ◽  
Totally Real ◽  
Totally Real Fields ◽  
Totally Real Field

We use results by Chenevier to interpolate the classical Jacquet–Langlands correspondence for Hilbert modular forms, which gives us an extension of Chenevier’s results to totally real fields. From this we obtain an isomorphism between eigenvarieties attached to Hilbert modular forms and those attached to modular forms on a totally definite quaternion algebra over a totally real field of even degree.

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

scholarly journals Eichler Maps and Hyperbolic Fourier Expansion

1970 ◽  
Vol 40 ◽  
pp. 173-192 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Fourier Expansion ◽  
Automorphic Forms ◽  
Hilbert Modular Forms ◽  
Ternary Quadratic Forms ◽  
Hilbert Modular ◽  
Lecture Notes

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

scholarly journals Siegel modular forms of degree two attached to Hilbert modular forms

2012 ◽  
Vol 132 (4) ◽  
pp. 543-564 ◽  
Author(s):  
Jennifer Johnson-Leung ◽  
Brooks Roberts
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Hilbert Modular Forms ◽  
Hilbert Modular
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