On the Curves Associated to Certain Rings of Automorphic Forms

2001 ◽  
Vol 53 (1) ◽  
pp. 98-121 ◽  
Author(s):  
Kamal Khuri-Makdisi
Keyword(s):  
Ad Hoc ◽  
Automorphic Forms ◽  
Complex Multiplication ◽  
Graded Rings ◽  
Projective Varieties ◽  
Hecke Operator ◽  
Complex Tori ◽  
Hecke Correspondences ◽  
Cm Points ◽  
Projective Lines

AbstractIn a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.

Arithmetic of Period Maps of Geometric Origin

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Hodge Structure ◽  
Complex Multiplication ◽  
Hodge Structures ◽  
Projective Varieties ◽  
Variation Of Hodge Structure ◽  
Cm Points ◽  
Geometric Origin ◽  
Isolated Points ◽  
Smooth Projective Varieties ◽  
Related Observation

This chapter considers some arithmetic aspects of period maps with a geometric origin. It focuses on the situation Φ‎ : S(ℂ) → Γ‎\D, where S parametrizes a family X → S of smooth, projective varieties defined over a number field k. The chapter recalls the notion of absolute Hodge classes (AH) and strongly absolute Hodge classes (SAH). The particular case when the Noether-Lefschetz locus consists of isolated points is alluded to in the discussion of complex multiplication Hodge structures (CM Hodge structures). A related observation is that one may formulate a variant of the “Grothendieck conjecture” in the setting of period maps and period domains. The chapter also describes a behavior of fields of definition under the period map, along with the existence and density of CM points in a motivic variation of Hodge structure.

A GEOMETRIC VERSION OF DYSON'S LEMMA FOR FACTORS OF ARBITRARY DIMENSION

2002 ◽  
Vol 13 (01) ◽  
pp. 43-65 ◽  
Author(s):  
MARKUS WESSLER
Keyword(s):  
Algebraic Geometry ◽  
Arbitrary Dimension ◽  
Product Theorem ◽  
Projective Varieties ◽  
Quantitative Version ◽  
Weak Positivity ◽  
Projective Lines ◽  
Smooth Projective Varieties ◽  
Geometric Analogue

This paper generalizes the geometric part of the Esnault–Viehweg paper on Dyson's Lemma for a product of projective lines. Using the method of weak positivity from algebraic geometry, we are able to study products of smooth projective varieties of arbitrary dimension and to prove a geometric analogue of Dyson's Lemma for this case. Our main result is in fact a quantitative version of Faltings' product theorem.

scholarly journals Twisted Gross–Zagier Theorems

2009 ◽  
Vol 61 (4) ◽  
pp. 828-887 ◽  
Author(s):  
Benjamin Howard
Keyword(s):  
Complex Multiplication ◽  
Modular Curves ◽  
Shimura Curve ◽  
Modular Curve ◽  
Hida Theory ◽  
Companion Article ◽  
Cm Points ◽  
Hida Families ◽  
Derivatives Of

Abstract.The theorems of Gross–Zagier and Zhang relate the Néron–Tate heights of complex multiplication points on the modular curve X0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-function. We extend these results to include certain CM points on modular curves of the form X (Ⲅ0(M ) ∩ Ⲅ1(S)) (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article “Central derivatives of L -functions in Hida families”,Math. Ann. 399(2007), 803–818.

scholarly journals Cohomologie d’intersection des variétés modulaires de Siegel, suite

2011 ◽  
Vol 147 (6) ◽  
pp. 1671-1740 ◽  
Author(s):  
Sophie Morel
Keyword(s):  
Fixed Point ◽  
Trace Formula ◽  
Compact Support ◽  
Automorphic Forms ◽  
Shimura Varieties ◽  
Academic Press ◽  
Test Functions ◽  
Hecke Operator ◽  
Frobenius Endomorphism ◽  
Modular Varieties

AbstractIn this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur’s invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur’s L2 trace formula and the fixed point formula of Morel [Complexes pondérés sur les compactifications de Baily–Borel. Le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61]. We ‘stabilize’ this last formula, i.e. express it as a sum of stable distributions on the general symplectic groups and its endoscopic groups, and obtain the formula conjectured by Kottwitz in [Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties and L-functions, Part I, Perspectives in Mathematics, vol. 10 (Academic Press, San Diego, CA, 1990), 161–209]. Applications of the results of this article have already been given by Kottwitz, assuming Arthur’s conjectures. Here, we give weaker unconditional applications in the cases of the groups GSp4 and GSp6.

scholarly journals A RECOLLEMENT APPROACH TO GEIGLE–LENZING WEIGHTED PROJECTIVE VARIETIES

2016 ◽  
Vol 226 ◽  
pp. 71-105 ◽  
Author(s):  
BORIS LERNER ◽  
STEFFEN OPPERMANN
Keyword(s):  
Algebraic Geometry ◽  
Abelian Category ◽  
New Method ◽  
Projective Varieties ◽  
Noncommutative Algebraic Geometry ◽  
Coherent Sheaves ◽  
Endomorphism Algebras ◽  
Projective Lines

We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle–Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties, and analyze the endomorphism algebras of their tilting bundles.

scholarly journals ANDRÉ–OORT CONJECTURE AND NONVANISHING OF CENTRAL -VALUES OVER HILBERT CLASS FIELDS

2016 ◽  
Vol 4 ◽  
Author(s):  
ASHAY A. BURUNGALE ◽  
HARUZO HIDA
Keyword(s):  
Modular Forms ◽  
Class Group ◽  
Complex Multiplication ◽  
Quadratic Extension ◽  
Real Field ◽  
Shimura Variety ◽  
Group Characters ◽  
Hilbert Modular ◽  
Cm Points ◽  
Definition Of

Let $F/\mathbf{Q}$ be a totally real field and $K/F$ a complex multiplication (CM) quadratic extension. Let $f$ be a cuspidal Hilbert modular new form over $F$. Let ${\it\lambda}$ be a Hecke character over $K$ such that the Rankin–Selberg convolution $f$ with the ${\it\theta}$-series associated with ${\it\lambda}$ is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg $L$-values $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$, as ${\it\chi}$ varies over the class group characters of $K$. Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters ${\it\chi}$ such that $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$ increases with the absolute value of the discriminant of $K$. We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg $L$-values. The Waldspurger formula plays an underlying role.

scholarly journals Seminormal graded rings and weakly normal projective varieties

1985 ◽  
Vol 8 (2) ◽  
pp. 231-240 ◽  
Author(s):  
John V. Leahy ◽  
Marie A. Vitulli
Keyword(s):  
Graded Rings ◽  
Projective Varieties ◽  
Normal Variety ◽  
Blowing Up ◽  
Weak Normality

This paper is concerned with the seminormality of reduced graded rings and the weak normality of projective varieties. One motivation for this investigation is the study of the procedure of blowing up a non-weakly normal variety along its conductor ideal.

scholarly journals F-THRESHOLDS OF GRADED RINGS

2016 ◽  
Vol 229 ◽  
pp. 141-168 ◽  
Author(s):  
ALESSANDRO DE STEFANI ◽  
LUIS NÚÑEZ-BETANCOURT
Keyword(s):  
Characteristic Zero ◽  
Projective Dimension ◽  
Graded Rings ◽  
Projective Varieties ◽  
Regular Sequences ◽  
Serre’S Condition ◽  
Regular Rings

The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.

scholarly journals Some remarks concerning Demazure’s construction of normal graded rings

1981 ◽  
Vol 83 ◽  
pp. 203-211 ◽  
Author(s):  
Keiichi Watanabe
Keyword(s):  
Finite Type ◽  
Homogeneous Element ◽  
Rational Coefficient ◽  
Graded Rings ◽  
Quotient Field ◽  
Graded Ring ◽  
Projective Varieties ◽  
Definition Of

In [1], Demazure showed a new way of constructing normal graded rings using the concept of “rational coefficient Weil divisors” of normal projective varieties and he showed, among other things, the followingTHEOREM ([1], 3.5). If R = ⊕n ≥ 0Rn is a normal graded ring of finite type over a field k and if T is a homogeneous element of degree 1 in the quotient field of R, then there exists unique divisor D ∈ Div (X, Q) (X = Proj (R)), such that for every n ≧ 0.(See (1.1) for the definition of

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