Abelian variety
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Mod $p$ points on Shimura varieties of abelian type Open
We show that the mod p points on a Shimura variety of abelian type with hyperspecial level have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contain…
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Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties Open
Given a quasiprojective variety $X$ with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus $X_{\\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main …
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Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn) Open
In recent papers we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situ…
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Categories of abelian varieties over finite fields, I: Abelian varieties over 𝔽<sub><i>p</i></sub> Open
We assign functorially a [math] -lattice with semisimple Frobenius action to each abelian variety over [math] . This establishes an equivalence of categories that describes abelian varieties over [math] avoiding [math] as an eigenvalue of …
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A converse to a theorem of Gross, Zagier, and Kolyvagin Open
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes and if the rank of $E(\mathbb{Q})$…
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Finiteness theorems for K3 surfaces and abelian varieties of CM type Open
We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational …
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Distinguished cycles on varieties with motive of abelian type and the Section Property Open
A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s sp…
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Remarks on motives of abelian type Open
A motive over a field $k$ is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over $k$. This paper contains three sections of independent interest. First, we show …
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The Betti map associated to a section of an abelian scheme Open
Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordin…
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INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE Open
Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$ . We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serr…
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Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields Open
We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S -integral points implies finiteness (resp. non-Zariski-density) of points over all $$\mathbb {Z}$$ -finitely generated integra…
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Algebraic cycles on a generalized Kummer variety Open
We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying abel…
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On good reduction of some K3 surfaces related to abelian surfaces Open
The Neron--Ogg--Safarevic criterion for abelian varieties tells that whether\nan abelian variety has good reduction or not can be determined from the Galois\naction on its l-adic etale cohomology. We prove an analogue of this criterion\nfo…
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A $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties Open
Mazur, Tate, and Teitelbaum gave a $p$-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties…
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Equidistribution in Families of Abelian Varieties and Uniformity Open
Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer $g \geq 2$, we pr…
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Strict supports of canonical measures and applications to the geometric Bogomolov conjecture Open
The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvariety. In the arithmetic setting over number fields, the Bogomolov conjecture for abelian varieties has already be…
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The basepoint-freeness threshold and syzygies of abelian varieties Open
We show how a natural constant introduced by Jiang and Pareschi for a\npolarized abelian variety encodes information about the syzygies of the section\nring of the polarization. As a particular case this gives a quick and\ncharacteristic-f…
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A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf Open
For an ordinary abelian variety X, $$F^e_*\mathcal {O}_X$$ is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and the Kodaira dimension of X is non-negative, t…
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Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces Open
We give several results concerning the connected component ${\\rm Aut}_X^0$ of\nthe automorphism scheme of a proper variety $X$ over a field, such as its\nbehaviour with respect to birational modifications, normalization, restrictions\nto …
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A special point problem of André-Pink-Zannier in the universal family of abelian varieties Open
The André-Pink-Zannier conjecture predicts that a subvariety of a mixed Shimura variety is weakly special if its intersection with the generalized Hecke orbit of a given point is Zariski dense.It is part of the Zilber-Pink conjecture.In th…
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Faltings heights of abelian varieties with complex multiplication Open
Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of specia…
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On the growth of Mordell–Weil ranks in $p$-adic Lie extensions Open
Let $p$ be an odd prime and $F_{\\infty}$ a $p$-adic Lie extension of a number\nfield $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction\nat every primes above $p$. Under various assumptions, we establish asymptotic\n…
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The Abelian-Nonabelian Correspondence for $I$-functions Open
We prove the abelian-nonabelian correspondence for quasimap $I$-functions. That is, if $Z$ is an affine l.c.i. variety with an action by a complex reductive group $G$, we prove an explicit formula relating the quasimap $I$-functions of the…
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Level structures on abelian varieties and Vojta’s conjecture Open
Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$ , there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abe…
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Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration Open
Let $X$ be an abelian scheme over a scheme $B$ . The Fourier–Mukai transform gives an equivalence between the derived category of $X$ and the derived category of the dual abelian scheme. We partially extend this to certain schemes $X$ over…
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On the distribution of rational points on ramified covers of abelian varieties Open
We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$ , where $A$ is an abelian va…
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The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao) Open
Given a point $ξ$ on a complex abelian variety $A$, its abelian logarithm can be expressed as a linear combination of the periods of $A$ with real coefficients, the Betti coordinates of $ξ$. When $(A, ξ)$ varies in an algebraic family, the…
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Perfectoid covers of abelian varieties Open
For an abelian variety $A$ over an algebraically closed non-archimedean field of residue characteristic $p$, we show that there exists a perfectoid space which is the tilde-limit of $\varprojlim_{[p]}A$. Our proof also works for the larger…
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Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato–Tate groups over a number field Open
Let A/Q be an abelian variety of dimension g = 1 that is isogenous over Q to Eg, where E is an elliptic curve. If E does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic Q-cur…
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Polarized endomorphisms of normal projective threefolds in arbitrary characteristic Open
Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1…