When is a function a Hilbert modular form?

2015 ◽  
Vol 47 (2) ◽  
pp. 285-289
Author(s):  
Peter Swinnerton-Dyer
Keyword(s):  
Modular Form ◽  
Hilbert Modular Form ◽  
Hilbert Modular

ARITHMETIC ON A QUINTIC THREEFOLD

2001 ◽  
Vol 12 (08) ◽  
pp. 943-972 ◽  
Author(s):  
CATERINA CONSANI ◽  
JASPER SCHOLTEN
Keyword(s):  
Modular Form ◽  
Theta Series ◽  
Continuous System ◽  
Galois Representation ◽  
Hilbert Modular Form ◽  
Galois Action ◽  
Hilbert Modular ◽  
Common Eigenvector ◽  
Quintic Threefold ◽  
Real Quadratic

This paper investigates some aspects of the arithmetic of a quintic threefold in Pr 4 with double points singularities. Particular emphasis is given to the study of the L-function of the Galois action ρ on the middle ℓ-adic cohomology. The main result of the paper is the proof of the existence of a Hilbert modular form of weight (2, 4) and conductor 30, on the real quadratic field [Formula: see text], whose associated (continuous system of) Galois representation(s) appears to be the most likely candidate to induce the scalar extension [Formula: see text]. The Hilbert modular form is interpreted as a common eigenvector of the Brandt matrices which describe the action of the Hecke operators on a space of theta series associated to the norm form of a quaternion algebra over [Formula: see text] and a related Eichler order.

Lambert series associated to Hilbert modular form

2021 ◽  
pp. 1-15
Author(s):  
Rishabh Agnihotri
Keyword(s):  
Asymptotic Expansion ◽  
Modular Form ◽  
Cusp Form ◽  
Riemann Zeta Function ◽  
Hilbert Modular Form ◽  
Lambert Series ◽  
Nontrivial Zeros ◽  
Hilbert Modular ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

scholarly journals On norm relations for Asai–Flach classes

2020 ◽  
Vol 16 (10) ◽  
pp. 2311-2377
Author(s):  
Giada Grossi
Keyword(s):  
Modular Form ◽  
Euler System ◽  
Real Field ◽  
Hilbert Modular Form ◽  
Hilbert Modular

We give a new proof of the norm relations for the Asai–Flach Euler system built by Lei–Loeffler–Zerbes. More precisely, we redefine Asai–Flach classes in the language used by Loeffler–Skinner–Zerbes for Lemma–Eisenstein classes and prove both the vertical and the tame norm relations using local zeta integrals. These Euler system norm relations for the Asai representation attached to a Hilbert modular form over a quadratic real field [Formula: see text] have been already proved by Lei–Loeffler–Zerbes for primes which are inert in [Formula: see text] and for split primes satisfying some assumption; with this technique we are able to remove it and prove tame norm relations for all inert and split primes.

scholarly journals On the equality case of the Ramanujan Conjecture for Hilbert modular forms

2019 ◽  
Vol 15 (10) ◽  
pp. 2107-2114
Author(s):  
Liubomir Chiriac
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Complex Multiplication ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Automorphic Representations ◽  
Equality Case ◽  
Hilbert Modular ◽  
Ramanujan Conjecture ◽  
Satake Parameters

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations [Formula: see text] on [Formula: see text] posits that [Formula: see text]. We prove that this inequality is strict if [Formula: see text] is generated by a Hilbert modular form of weight two, with complex multiplication, and [Formula: see text] is a finite place of degree one. Equivalently, the Satake parameters of [Formula: see text] are necessarily distinct. We also give examples where the equality case does occur for places [Formula: see text] of degree two.

scholarly journals Hilbert modular forms and p-adic Hodge theory

2009 ◽  
Vol 145 (5) ◽  
pp. 1081-1113 ◽  
Author(s):  
Takeshi Saito
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Galois Representation ◽  
Hodge Theory ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Langlands Correspondence ◽  
Hilbert Modular ◽  
Elliptic Modular Form ◽  
Local Langlands Correspondence

AbstractFor the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

ON NONCRITICAL GALOIS REPRESENTATIONS

2021 ◽  
pp. 1-38
Author(s):  
Bingyong Xie
Keyword(s):  
Modular Form ◽  
Galois Representations ◽  
Galois Representation ◽  
Hilbert Modular Form ◽  
Hilbert Modular

Abstract We propose a conjecture that the Galois representation attached to every Hilbert modular form is noncritical and prove it under certain conditions. Under the same condition we prove Chida, Mok and Park’s conjecture that Fontaine-Mazur L-invariant and Teitelbaum-type L-invariant coincide with each other.

scholarly journals On the number of Fourier coefficients that determine a Hilbert modular form

2002 ◽  
Vol 130 (9) ◽  
pp. 2497-2502 ◽  
Author(s):  
Srinath Baba ◽  
Kalyan Chakraborty ◽  
Yiannis N. Petridis
Keyword(s):  
Modular Form ◽  
Fourier Coefficients ◽  
Hilbert Modular Form ◽  
Hilbert Modular

scholarly journals An analogue of a conjecture of Sato and Tate for a Hilbert modular form

1975 ◽  
Vol 16 (2) ◽  
pp. 69-87 ◽  
Author(s):  
H. L. Resnikoff ◽  
R. L. Saldaña
Keyword(s):  
Modular Form ◽  
Elliptic Curve ◽  
Density Function ◽  
Number Field ◽  
Hilbert Modular Form ◽  
Hilbert Modular ◽  
Complex Multiplications ◽  
Image Position

If k denotes a number field and εm is the product of an elliptic curve ε with itself m times over k, then for each prime π where ε has non-degenerate reduction, the zeta factor ζ(επ'S) can be expressed asWhere |π| denotes the norm of π. It is a consequence of a conjecture of Tate [16] that if ε does not have complex multiplications, then the numbers are distributed according to the density functionthat is, the density of the set of primes π such that – is

scholarly journals Congruences for convolutions of Hilbert modular forms

2012 ◽  
Vol 153 (3) ◽  
pp. 471-487 ◽  
Author(s):  
THOMAS WARD
Keyword(s):  
Modular Form ◽  
Finite Order ◽  
Modular Forms ◽  
Iwasawa Theory ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Theta Lift ◽  
Hilbert Modular ◽  
Rankin Convolution ◽  
Tate Curve

AbstractLet f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution L-values L(f,g,s), where g is a theta-lift modular form corresponding to a finite-order character. We prove weak forms of Kato's ‘false Tate curve’ congruences for these values, of the form predicted by conjectures in non-commmutative Iwasawa theory.

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