Frank Calegari
Continuing on from recent results of Brumer-Kramer and of Schoof, we show that there exist non-zero semistable Abelian varieties over Z[1/N], with N squarefree, if and only if N is not in the set {1,2,3,5,6,7,10,13}. Our results are contingent on the GRH discriminant bounds of Odlyzko.
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We show that there do not exist semistable varietes defined over the rationals with good reduction outside one prime p if p = 2, 3, 5 or 7.
We prove the Iwasawa main conjecture over the arithmetic $\mathbb{Z}_p$-extension for semistable abelian varieties over function fields of characteristic $p>0$.
We completely solve a problem of S. Zhang about the positivity of a normalized height on the moduli space of semistable varieties of given degree and given dimension.
Semistable abelian varieties over $\mathbb{Q}$ with bad reduction at 19 only: an overview of the Fontaine--Schoof strategy
本文概述了Fontaine首创并由Schoof改进的分类具有指定坏约化的阿贝尔簇的策略，并给出了新结果：分类仅在19处有坏约化的半稳定阿贝尔簇。
We prove that for abelian varieties with semistable ordinary reduction the p-adic Mazur-Tate height pairing is induced by the unit root splitting of the Hodge filtration on the first deRham cohomology.
Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(μ_\ell)$ be the cyclotomic field generated by the group of $\ell^{th}$-roots of unity. We study the varieties $A$ for which $H={\rm Gal(L/F)}$ is "small" in the sense that $H$ is an $\ell$-group or, more generally, that $H$ is nilpotent. We show that if $\ell=2$ or 3 and $H$ is nilpotent then the reduction of $A$ at $p$ is totally toroidal, so its conductor is $p^{\dim A}$. T…
We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this translates into a purely algebraic problem concerning the number of $H$-invariant points on a quotient of $C_n$-lattices $Λ/eΛ'$ for varying subgroups $H$ of $C_n$ and integers $e$. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions (corresponding to varying $e$) and one that computes Tamagawa numbers up to rational squares in general extensions. As an application…
Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\barρ_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of $G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the $\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if image of $\barρ_{A,\ell}$ contains a transvection then $\barρ_{A,\ell}$ is either reducible or surjective. With the help of this we study surjectivity of $\barρ_{A,\ell}$ for semistable princi…
We define an overconvergent version of the Hyodo-Kato complex for semistable varieties $Y$ over perfect fields of positive characteristic, and prove that its hypercohomology tensored with $\mathbb{Q}$ recovers the log-rigid cohomology when $Y$ is quasi-projective. We then describe the monodromy operator using the overconvergent Hyodo-Kato complex. Finally, we show that overconvergent Hyodo-Kato cohomology agrees with log-crystalline cohomology in the projective semistable case.
We prove that for N=6 and N=10, there do not exist any non-zero semistable abelian varieties over Q with good reduction outside primes dividing N. Our results are contingent on the GRH discriminant bounds of Odlyzko. Combined with recent results of Brumer--Kramer and of Schoof, this result is best possible: if N is squarefree, there exists a non-zero semistable abelian variety over Q with good reduction outside primes dividing N precisely when N is not in the set {1,2,3,5,6,7,10,13}.
In this paper we prove a comparison theorem between the category of certain modules with integrable connection on the complement of a normal crossing divisor of the generic fiber of a proper semistable variety over a DVR and the category of certain log overconvergent isocystrals on the special fiber of the same open.
This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequ…
In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This in particular implies Grothendieck's conjecture on the perfectness of his pairing between the Neron component groups of an abelian variety and its dual. The point is that our formulation is well-suited with Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.
We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over $p$-adic rings extends uniquely to a cohomology theory for varieties over $p$-adic fields that satisfies $h$-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to étale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the étale side. In relative dimension zero we recover the potentiall…
A log-motivic cohomology for semistable varieties and its $p$-adic deformation theory
本文提出了一种半稳定概形的log-动机上同调理论，并探讨了其p进变形理论，通过研究Hyodo-Kato类与Hodge滤波的关系，推广了Bloch-Esnault-Kerz的结论。
Let $p$ be a prime, let $K$ be a discretely valued extension of $\mathbb{Q}_p$, and let $A_{K}$ be an abelian $K$-variety with semistable reduction. Extending work by Kim and Marshall from the case where $p>2$ and $K/\mathbb{Q}_p$ is unramified, we prove an $l=p$ complement of a Galois cohomological formula of Grothendieck for the $l$-primary part of the Néron component group of $A_{K}$. Our proof involves constructing, for each $m\in \mathbb{Z}_{\geq 0}$, a finite flat $\mathscr{O}_K$-group scheme with generic fiber equal to the maximal 1-crystalline submodule of $A_{K}[p^{m}]$. As a corollar…
We study the arithmetic of division fields of semistable abelian varieties A over the rationals. The Galois group of the 2-division field of A is analyzed when the conductor is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in "Paramodular abelian varieties of odd conductor," arXiv:1004.4699.
We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, embeds into the reduction modulo p of a semistable Galois representation with Hodge-Tate weights in the expected range (at least after semisimplifying, in the case of the cohomological degree > 1). We prove refinements with descent data, and we apply these results to the cohomology of unitary Shimura varieties, deducing vanishing results and applications to the weight part of Serre's conjecture.
We introduce a qualitative conjecture, in the spirit of Campana, to the effect that certain subsets of rational points on a variety over a number field, or a Deligne-Mumford stack over a ring of S-integers, cannot be Zariski dense. The conjecture interpolates, in a way that we make precise, between Lang's conjecture for rational points on varieties of general type over number fields, and the conjecture of Lang and Vojta that asserts that S-integral points on a variety of logarithmic general type are not Zariski-dense. We show our conjecture follows from Vojta's conjecture. Assuming our conject…
Around $\ell$-independence
本文研究有限域与等特征局部域上簇的ℓ-独立性，通过“展开”结果推导半稳定情形的弱ℓ-独立性，还得到Clemens–Schmid正合列存在性等推论。
Strongly compatible systems associated to semistable abelian varieties
本文证明了数域上阿贝尔簇的$\ell$-adic上同调作用在有限扩张后可给出取值于Mumford--Tate群的强相容表示系，并将Weil--Deligne表示的$\ell$独立性从良约化情形推广到半稳定约化情形。
For an optimal modular parametrization $J_0(n) \twoheadrightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^0(E, Ω^1)$. Multiple authors generalized his conjecture to higher dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic etale and de Rham coho…
In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.
Rational points and Brauer--Manin obstruction on Shimura varieties of level one classifying abelian varieties with quaternionic multiplication
该研究探讨具有四元数乘法的阿贝尔簇对应的一级志村簇的有理点问题，证明这类志村簇极少存在有理点，为数论中志村簇有理点分布研究提供了结论。
We prove control theorems for abelian varieties over function fields.
Sharp bounds on the height of K-semistable Fano varieties I, the toric case
受Fujita结果启发，本文猜想整数射影空间在K-半稳定度量算术Fano簇中高度最大，并在n≤6时证明了环面Fano簇的该猜想，给出了Donaldson环面不变量与Odaka模高度的尖锐下界。
We study monodromy action on abelian varieties satisfying certain bad reduction conditions. These conditions allow us to get some control over the Galois image. As a consequence we verify the Mumford--Tate conjecture for such abelian varieties.
Rational points on Shimura varieties classifying abelian varieties with quaternionic multiplication
本文研究具有四元数乘法的阿贝尔簇的适当水平结构诱导的特征，并证明参数化它们的Shimura簇很少有有理点。
Semistable Reduction Theorem for Overconvergent $F$-isocrystals over Laurent Series Fields
本文证明了在劳伦茨级数域上，针对拉兹达和帕尔引入的过收敛F-等晶体的半稳定约简定理，并应用该结果证明了带有紧支撑的$\mathcal{E}^†_K$-valued刚性上同调的有限维性。
We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.
In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.
In this paper, we study a conjecture of Rasmussen and Tamagawa, on the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion. Our result is related with abelian varieties which have complex multiplication over their fields of definition.
We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.
We prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special.
We provide a characterization of almost ordinary abelian varieties over finite fields, and use this characterization to provide lower bounds for the sizes of some almost ordinary isogeny classes.
Sharp bounds on the height of K-semistable Fano varieties II, the log case
本文将关于 K-半稳定算术 Fano 簇高度上界的猜想推广到对数情形，并据此证明整数射影空间中对角超曲面的高度以 Fubini-Study 射影空间为最大。
We show that certain abelian varieties over $\Q$ with bad reduction at one prime only are modular by using methods based on the tables of Odlyzko and class field theory.
In this paper we determine the number of isomorphism classes of superspecial abelian varieties $A$ over the prime field $\Fp$ such that the relative Frobenius morphism $π_A$ satisfying $π_A^2=-p$.
We study a certain class of simple abelian varieties of type $\mathrm{IV}$ (in Albert's classification) over number fields with Mumford-Tate groups of type $A$. In particular, we show that such abelian varieties have ordinary reduction away from a set of places of Dirichlet density zero, thus confirming a special case of a broader conjecture of Serre's. We also study the splitting types and Newton polygons of the reductions of the abelian varieties of this type with small dimension (nine).
We characterize the abelian varieties arising as absolutely simple factors of GL2-type varieties over a number field k. In order to obtain this result, we study a wider class of abelian varieties: the k-varieties A/k satisfying that $\End_k^0(A)$ is a maximal subfield of $\End_k^0(A)$. We call them Ribet-Pyle varieties over k. We see that every Ribet-Pyle variety over k is isogenous over $\bar k$ to a power of an abelian k-variety and, conversely, that every abelian k-variety occurs as the absolutely simple factor of some Ribet-Pyle variety over k. We deduce from this correspondence a precise …
We prove that Shimura varieties of abelian type with infinite level at $p$ are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at $p$ are also perfectoid.
Introduction to Dieudonné modules and supersingular abelian varieties revisited
本文介绍了Dieudonné模块和奇超阿贝尔簇，给出了奇超椭圆曲线乘积唯一性和Oort定理的简单证明。
If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism from A to B defined over an algebraic closure of K.
We discuss heuristic asymptotic formulae for the number of pairing-friendly abelian varieties over prime fields, generalizing previous work of one of the authors arXiv:math1107.0307
In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k [\![ t ]\!]$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter example to show that the variational Tate conject…
Finiteness of Heights in Isogeny Classes of Motives with Semistable Reduction
本文利用积分 $p$-adic Hodge 理论推广了阿贝尔簇的 Faltings 高度，并在假设 adelic Mumford-Tate 猜想下，证明了 motive 同构类中高度的有限性，扩展了 Kisin 和 Mocz 的相关结果。
We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties.
The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties.
We study (weak) log abelian varieties with constant degeneration in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In particular, we prove the dual short exact sequence for isogenies, Poincaré complete reducibility theorem for log abelian varieties, and the semi-simplicity of the endomorphism algebras of log abelian varieties.
Hilbert irreducibility for abelian varieties over function fields of characteristic zero
该研究证明了特征零函数域上阿贝尔簇的希尔伯特不可约定理，完善了相关数论与代数几何理论。
In this paper we study super-isolated abelian varieties, that is, abelian varieties over finite fields whose isogeny class contains a single isomorphism class. The goal of this paper is to (1) characterize whether a product of super-isolated varieties is super-isolated, and (2) characterize which super-isolated abelian varieties admit principal polarizations, and how many up to polarized isomorphisms.
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
In this paper we discuss multiplicative relations between eigenvalues of Frobenius endomorphism of abelian varieties of small dimension over finite fields.
We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.
We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial endomorphisms.
Analogy between Anderson T-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: 1. A description of abelian varieties with MIQF of dimension $r$ and signature $(n, r-n)$ in terms of "lattices" of dimension $r$ in $\Bbb C^n$; 2. A construction of exterior powers of abelian varieties with MIQF having $n=1$.
In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields. Here we rework the proofs so that they apply to a single abelian variety. As a consequence, we prove (unconditionally) that the Tate and standard conjectures are true for many abelian varieties over finite fields, including abelian varieties for which the algebra of Tate classes is not generated by divisor classes.
We study in this paper some criterions to get polarized morphisms between abelian varieties. We deduce explicit dynamical systems with particular intersection properties.
We prove various characterizations of the period torsor of abelian varieties. This is the submitted version.
We prove that torsion codimension 2 algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension 2 on supersingular abelian varieties over the algebraic closure of finite fields.
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