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.

scholarly journals Shimura curves in the Prym locus

2019 ◽  
Vol 21 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Elisabetta Colombo ◽  
Paola Frediani ◽  
Alessandro Ghigi ◽  
Matteo Penegini
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Shimura Curves ◽  
Simple Criterion ◽  
One Dimensional ◽  
Fixed Group ◽  
The Family ◽  
Base Curve ◽  
Double Covers ◽  
Prym Map

We study Shimura curves of PEL type in [Formula: see text] generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases, we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is one-dimensional and the quotient of the base curve by the group is [Formula: see text]. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples obtained in this way up to genus 28. We obtain 43 Shimura curves contained in the unramified Prym locus and 9 families contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.

scholarly journals Shimura subvarieties in the Prym locus of ramified Galois coverings

2021 ◽  
Author(s):  
Gian Paolo Grosselli ◽  
Abolfazl Mohajer
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Fixed Group ◽  
Galois Coverings ◽  
Base Curve ◽  
Prym Map

AbstractWe study Shimura (special) subvarieties in the moduli space $$A_{p,D}$$ A p , D of complex abelian varieties of dimension p and polarization type D. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $${{\mathbb {P}}}^1$$ P 1 . We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

scholarly journals A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

2019 ◽  
Vol 31 (5) ◽  
pp. 1317-1330
Author(s):  
Russell Ricks
Keyword(s):  
Symmetric Space ◽  
Group Action ◽  
Spherical Building ◽  
Rigidity Result ◽  
One Dimensional ◽  
Higher Rank ◽  
Geodesically Complete ◽  
Euclidean Building ◽  
Alternate Condition

AbstractWe prove the following rank rigidity result for proper {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of {\partial X} equals π and Γ does not act minimally on {\partial X}, then {\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of {\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

scholarly journals Index conditions and cup-product maps on Abelian varieties

2014 ◽  
Vol 25 (04) ◽  
pp. 1450036 ◽  
Author(s):  
Nathan Grieve
Keyword(s):  
Asymptotic Behavior ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
General Setting ◽  
Index Theorem ◽  
Line Bundles ◽  
One Dimensional ◽  
Cup Product ◽  
Product Maps

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behavior of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one-dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.

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.

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.

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