scholarly journals Rational points on abelian varieties over function fields and Prym varieties

2020 ◽  
Author(s):  
Abolfazl Mohajer
Keyword(s):  
Structure Theorem ◽  
Function Fields ◽  
Abelian Varieties ◽  
High Rank ◽  
Prym Variety ◽  
Projective Varieties ◽  
Prym Varieties ◽  
Weil Group ◽  
Galois Covers ◽  
Smooth Projective Varieties

AbstractIn this paper, using a generalization of the notion of Prym variety for covers of projective varieties, we prove a structure theorem for the Mordell–Weil group of abelian varieties over function fields that are twists of abelian varieties by Galois covers of smooth projective varieties. In particular, the results we obtain contribute to the construction of Jacobians of high rank.

scholarly journals Abelian Varieties as Automorphism Groups of Smooth Projective Varieties

2018 ◽  
Vol 2020 (7) ◽  
pp. 1942-1956
Author(s):  
Davide Lombardo ◽  
Andrea Maffei
Keyword(s):  
Automorphism Group ◽  
Projective Variety ◽  
Automorphism Groups ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Projective Varieties ◽  
Smooth Projective Varieties

Abstract We determine which complex abelian varieties can be realized as the automorphism group of a smooth projective variety.

scholarly journals Derived equivalent threefolds, algebraic representatives, and the coniveau filtration

2018 ◽  
Vol 167 (01) ◽  
pp. 123-131 ◽  
Author(s):  
JEFFREY D. ACHTER ◽  
SEBASTIAN CASALAINA-MARTIN ◽  
CHARLES VIAL
Keyword(s):  
Abelian Varieties ◽  
Projective Varieties ◽  
Chow Motives ◽  
Smooth Projective Varieties ◽  
Coniveau Filtration

AbstractA conjecture of Orlov predicts that derived equivalent smooth projective varieties over a field have isomorphic Chow motives. The conjecture is known for curves, and was recently observed for surfaces by Huybrechts. In this paper we focus on threefolds over perfect fields, and unconditionally secure results, which are implied by Orlov's conjecture, concerning the geometric coniveau filtration, and abelian varieties attached to smooth projective varieties.

scholarly journals Generic Torelli theorem for Prym varieties of ramified coverings

2012 ◽  
Vol 148 (4) ◽  
pp. 1147-1170 ◽  
Author(s):  
Valeria Ornella Marcucci ◽  
Gian Pietro Pirola
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Branch Points ◽  
Prym Variety ◽  
Prym Varieties ◽  
Torelli Theorem ◽  
Image Position ◽  
Prym Map ◽  
Double Coverings ◽  
Ramified Coverings

AbstractWe consider the Prym map from the space of double coverings of a curve of genus gwithrbranch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2is generically injective ifWe also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

scholarly journals On Mordell–Weil groups of Jacobians over function fields

2012 ◽  
Vol 12 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Douglas Ulmer
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Abelian Varieties ◽  
High Rank ◽  
Arbitrary Field ◽  
New Construction

AbstractWe study the arithmetic of abelian varieties over $K= k(t)$ where $k$ is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over ${ \overline{ \mathbb{F} } }_{p} (t)$ and a new construction of elliptic curves with moderately high rank over $ \mathbb{C} (t)$.

scholarly journals On the arithmetic of abelian varieties

2020 ◽  
Vol 2020 (762) ◽  
pp. 1-33
Author(s):  
Mohamed Saïdi ◽  
Akio Tamagawa
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Galois Cohomology ◽  
Rational Points ◽  
Selmer Groups ◽  
Selmer Group ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Natural Objects ◽  
Weil Group

AbstractWe prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects “discrete Selmer groups” and “discrete Shafarevich–Tate groups”, and prove that they are finitely generated {\mathbb{Z}}-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich–Tate group vanishes and the discrete Selmer group coincides with the Mordell–Weil group. One of the key ingredients to prove these results is a new specialisation theorem for first Galois cohomology groups, which generalises Néron’s specialisation theorem for rational points of abelian varieties.

scholarly journals ON VARIETIES WITH TRIVIAL TANGENT BUNDLE IN CHARACTERISTIC 0$" height="12pt">

2019 ◽  
pp. 1-17
Author(s):  
KIRTI JOSHI
Keyword(s):  
Abelian Variety ◽  
Tangent Bundle ◽  
Abelian Varieties ◽  
Projective Varieties ◽  
Crystalline Cohomology ◽  
Smooth Projective Varieties ◽  
Tangent Bundles ◽  
Torsion Free ◽  
Projective Surfaces

In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$ . Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.

Non-communtative Iwasawa main conjecture

2020 ◽  
Vol 16 (09) ◽  
pp. 2041-2094
Author(s):  
Malte Witte
Keyword(s):  
Functional Equation ◽  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Main Conjecture

We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for [Formula: see text]-adic representations of the Galois group of a function field of characteristic [Formula: see text]. We also prove a functional equation for the resulting non-commutative [Formula: see text]-functions. As corollaries, we obtain non-commutative generalizations of the main conjecture for Picard-[Formula: see text]-motives of Greither and Popescu and a main conjecture for abelian varieties over function fields in precise analogy to the [Formula: see text] main conjecture of Coates, Fukaya, Kato, Sujatha and Venjakob.

scholarly journals The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

2016 ◽  
Vol 112 (6) ◽  
pp. 1040-1058 ◽  
Author(s):  
King Fai Lai ◽  
Ignazio Longhi ◽  
Ki-Seng Tan ◽  
Fabien Trihan
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Main Conjecture
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