Trace of the Generating Series

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Eisenstein Series ◽  
Abelian Varieties ◽  
Discrete Series ◽  
Theta Series ◽  
Shimura Curves ◽  
Trace Identity ◽  
Shimura Curve ◽  
Schwartz Functions ◽  
Generating Series ◽  
New Space

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.

Mordell-Weil Groups and Generating Series

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Kernel Function ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Generating Series ◽  
Analytic Kernel

This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before introducing a theorem, which is an identity between the analytic kernel and the geometric kernel. It then defines the generating series and uses it to describe the geometric kernel. It also presents a theorem, which is an identity formulated in terms of projectors, and reviews some basic notations and results on Shimura curves. Other topics covered include the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves, normalization of the geometric kernel, and the analytic kernel function. The chapter concludes with an analysis of the kernel identity implied in the first theorem.

The Gross-Zagier Formula on Shimura Curves

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-wu Zhang ◽  
Wei Zhang
Keyword(s):  
Diophantine Equations ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Heegner Points ◽  
Quaternion Algebras ◽  
Complete Proof ◽  
Weil Representations ◽  
Arakelov Theory ◽  
Generating Series ◽  
New Applications

This comprehensive account of the Gross–Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross–Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross–Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross–Zagier formula is reduced to local formulas. This book will be of great use to students wishing to enter this area and to those already working in it.

scholarly journals Boundary behaviour of special cohomology classes arising from the Weil representation

2012 ◽  
Vol 12 (3) ◽  
pp. 571-634 ◽  
Author(s):  
Jens Funke ◽  
John Millson
Keyword(s):  
Modular Forms ◽  
Symmetric Spaces ◽  
Theta Functions ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Boundary Behaviour ◽  
Orthogonal Groups ◽  
Local Coefficients ◽  
Generating Series ◽  
Vector Valued

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

scholarly journals Diagrams in the mod p cohomology of Shimura curves

2021 ◽  
Vol 157 (8) ◽  
pp. 1653-1723
Author(s):  
Andrea Dotto ◽  
Daniel Le
Keyword(s):  
Shimura Curves ◽  
Shimura Curve ◽  
Langlands Program ◽  
Mod P

Abstract We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$ . Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$ . If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$ .

scholarly journals The -adic Gross–Zagier formula on Shimura curves

2017 ◽  
Vol 153 (10) ◽  
pp. 1987-2074 ◽  
Author(s):  
Daniel Disegni
Keyword(s):  
General Formula ◽  
Analytic Functions ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Root Number ◽  
Heegner Points ◽  
Semistable Reduction

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a $p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

ON THE THETA SERIES ATTACHED TO $D_{m}^{+}$–LATTICES

2006 ◽  
Vol 02 (01) ◽  
pp. 1-5 ◽  
Author(s):  
WINFRIED KOHNEN ◽  
RICCARDO SALVATI MANNI
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Eisenstein Series ◽  
Theta Series

We show that the theta series attached to the [Formula: see text]-lattice for any positive integer divisible by 8 can be explicitly expressed as a finite rational linear combination of products of two Eisenstein series.

scholarly journals The Saito–Kurokawa lift and Siegel’s theta series

2017 ◽  
Vol 13 (07) ◽  
pp. 1679-1693
Author(s):  
Roland Matthes
Keyword(s):  
Spectral Analysis ◽  
Eisenstein Series ◽  
Short Proof ◽  
Theta Series ◽  
Converse Theorem ◽  
Selberg Integral ◽  
Real Analytic

The aim of this paper is to give another short proof of the Saito–Kurokawa lift based on a converse theorem of Imai as was already done by Duke and Imamoglu. In contrast to their proof we avoid spectral analysis but use a real analytic Eisenstein series in a suitable Rankin–Selberg integral involving Siegel’s theta series.

scholarly journals Unitary cycles on Shimura curves and the Shimura lift II

2014 ◽  
Vol 150 (12) ◽  
pp. 1963-2002
Author(s):  
Siddarth Sankaran
Keyword(s):  
Modular Form ◽  
Recent Work ◽  
Chow Groups ◽  
Shimura Varieties ◽  
Shimura Curves ◽  
Shimura Lift ◽  
Generating Series

AbstractWe consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the$q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.

scholarly journals Shimura curves in the Prym loci of ramified double covers

2021 ◽  
Author(s):  
Paola Frediani ◽  
Gian Paolo Grosselli
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
One Dimensional ◽  
Fixed Group ◽  
Base Curve ◽  
Double Covers ◽  
Ramification Points

We study Shimura curves of PEL type in the space of polarized abelian varieties [Formula: see text] generically contained in the ramified Prym locus. We generalize to ramified double covers, the construction done in [E. Colombo, P. Frediani, A. Ghigi and M. Penegini, Shimura curves in the Prym locus, Commun. Contemp. Math. 21(2) (2019) 1850009] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is [Formula: see text]. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci.

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