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.

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 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.

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 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 Eisenstein series in the Kohnen plus space for Hilbert modular forms

2016 ◽  
Vol 12 (03) ◽  
pp. 691-723 ◽  
Author(s):  
Ren-He Su
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Hilbert Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
The One ◽  
Totally Real

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

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 On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

2013 ◽  
Vol 65 (2) ◽  
pp. 403-466
Author(s):  
Jeanine Van Order
Keyword(s):  
Modular Forms ◽  
Iwasawa Theory ◽  
The Other ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Totally Real ◽  
Totally Real Fields

AbstractWe construct a bipartite Euler systemin the sense ofHoward forHilbertmodular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies inmany cases one divisibility of the associated dihedral or anticyclotomicmain conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

scholarly journals Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case

2014 ◽  
Vol 2 ◽  
Author(s):  
PAYMAN L. KASSAEI ◽  
SHU SASAKI ◽  
YICHAO TIAN
Keyword(s):  
Analytic Continuation ◽  
General Result ◽  
Modular Forms ◽  
Galois Representations ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Weight One ◽  
Totally Real ◽  
Totally Real Fields ◽  
Finite Slope

AbstractWe extend the modularity lifting result of P. Kassaei (‘Modularity lifting in parallel weight one’,J. Amer. Math. Soc.26 (1) (2013), 199–225) to allow Galois representations with some ramification at $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$. We also prove modularity mod 5 of certain Galois representations. We use these results to prove new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent $p$-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in P. Kassaei (‘Modularity lifting in parallel weight one’, J. Amer. Math. Soc.26 (1) (2013), 199–225).

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