Mixed Ax–Schanuel for the universal abelian varieties and some applications

Ziyang Gao

doi:10.1112/s0010437x20007447

Mixed Ax–Schanuel for the universal abelian varieties and some applications

Compositio Mathematica ◽

10.1112/s0010437x20007447 ◽

2020 ◽

Vol 156 (11) ◽

pp. 2263-2297

Author(s):

Ziyang Gao

Keyword(s):

Abelian Varieties ◽

Shimura Variety ◽

Generic Rank

In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.

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Applications of the hyperbolic Ax–Schanuel conjecture

Compositio Mathematica ◽

10.1112/s0010437x1800725x ◽

2018 ◽

Vol 154 (9) ◽

pp. 1843-1888 ◽

Cited By ~ 4

Author(s):

Christopher Daw ◽

Jinbo Ren

Keyword(s):

Abelian Varieties ◽

Shimura Varieties ◽

Shimura Variety ◽

Point Counting ◽

Counting Problem

In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the$j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax–Schanuel conjecture. In this article, we show that the hyperbolic Ax–Schanuel conjecture can be used to reduce the Zilber–Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila–Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila–Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber–Pink conjecture for curves in abelian varieties.

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Analytic Theory of Abelian Varieties

10.1017/cbo9780511662621 ◽

1974 ◽

Cited By ~ 16

Author(s):

H. P. F. Swinnerton-Dyer

Keyword(s):

Abelian Varieties ◽

Analytic Theory

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System of generic groups in mealybugs (Homoptera: Coccinea: Pseudococcidae)

Zoosystematica Rossica ◽

10.31610/zsr/2015.24.2.236 ◽

2015 ◽

Vol 24 (2) ◽

pp. 236-259 ◽

Cited By ~ 10

Author(s):

I.A. Gavrilov-Zimin

Keyword(s):

Taxonomic Revision ◽

Generic Rank ◽

The World ◽

Generic Groups ◽

World Fauna

The paper provides a brief conspectus of the system of morphological generic groups, elaborated earlier by the author basing on the total taxonomic revision of Palaearctic mealybugs. Here the system is complemented by the analysis of all 249 genera of the world fauna. Borders of two generic groups are reconsidered and two else groups (with mainly Oriental and Australasian genera) are included in the system. Main taxonomic characters of generic rank are discussed and illustrated.

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Aspidisca beringiana sp. nov. and Simbiodisca subgen. nov. (Ciliophora, Euplotida), a symbiont of terebellid polychaetes in the Bering Sea

Zoosystematica Rossica ◽

10.31610/zsr/2009.18.2.184 ◽

2009 ◽

Vol 18 (2) ◽

pp. 184-190

Author(s):

A.W. Jankowski

Keyword(s):

Bering Sea ◽

Tidal Zone ◽

Generic Rank ◽

New Subgenus ◽

Bering Island ◽

The Bering Sea

Terebellids in tidal zone of the Bering Island bear three new symbionts - rhabdophryid suctorians, peritrichs with small rosette-like colonies and aspidiscid hypotrich with a long peristome parallel to left body margin. This is the main feature of a new subgenus of the genus Aspidisca, named Simbiodisca. It may deserve the full generic rank if the use of protargol silvering method will not reveal any upper left rudiment of the peristomal membranelles.

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On Abelian Varieties with Complex Multiplication as Factors of the Jacobians of Shimura Curves

American Journal of Mathematics ◽

10.2307/2374049 ◽

1981 ◽

Vol 103 (4) ◽

pp. 727 ◽

Cited By ~ 15

Author(s):

Haruzo Hida

Keyword(s):

Abelian Varieties ◽

Complex Multiplication ◽

Shimura Curves

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The larva of Psychoda (Philosepedon) humeralis Meigen (Diptera, Psychodidae)

Parasitology ◽

10.1017/s0031182000072620 ◽

1963 ◽

Vol 53 (1-2) ◽

pp. 155-156

Author(s):

Kenneth G. V. Smith ◽

L. W. Grensted

Keyword(s):

Larval Stage ◽

Sensu Stricto ◽

Present Account ◽

Generic Rank ◽

Detailed Treatment

Satchell (1947) described and keyed the larvae of 14 of the 19 British species of Psychoda, but this study did not include P. humeralis Mg., presumably because the author accorded generic rank to Philosepedon and Threticus which would put them outside his study of Psychoda sensu stricto. The larva of P. humeralis has been briefly described by Spärck (1920), but his figures are rather crude. A detailed treatment of the larval head is given by Anthon (1943). The present account is offered to facilitate identification of this species in the larval stage when used in conjunction with Satchell's comprehensive paper.

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