Michael Larsen, René Schoof
If A/K is an abelian variety over a number field and P and Q are rational points, the original support conjecture asserted that if the order of Q (mod p) divides the order of P (mod p) for almost all primes p of K, then Q is obtained from P by applying an endomorphism of A. This is now known to be untrue. In this note we prove that it is not even true modulo the torsion of A.
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A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety
研究p进对数下阿贝尔簇非 torsion 点非消失问题，主要针对GL₂型阿贝尔簇及其与希格纳点相关的情形，采用p进解析子群定理作为主要工具。
This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a con…
Refined obstructions to local-global principles for 0-cycles
本文引入新的精细化障碍，用于研究数字段上代数簇0-循环的局部-全局原理，通过Tate-Shafarevich群的有限性，探讨广义Kummer簇和双椭圆面的Hasse原理和弱逼近，并回答Zhang关于Brauer-Manin和连通降阶障碍关系的问题。
Let A be an abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k). Gajda in 2002 asked whether it is true that the point P belongs to X if and only if the point (P mod p) belongs to (X mod p) for all but finitely many primes p of k. We answer negatively to Gajda's question.
We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field. We remove the simplicity assumption in the principally polarized case by giving a refined version of…
We propose a refined version of the Beilinson-Bloch conjecture for the motive associated with a modular form of even weight. This conjecture relates the dimension of the image of the relevant p-adic Abel-Jacobi map to certain combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the direction of the conjecture and, in doing so, obtain higher weight analogues of results for elliptic curves due to Darmon.
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。
Let A be an abelian variety defined over a number field K and let P and Q be points in A(K) satisfying the following condition: for all but finitely many primes p of K, the order of (Q mod p) divides the order of (P mod p). Larsen proved that there exists a positive integer c such that cQ is in the End_K(A)-module generated by P. We study the minimal value of c and construct some refined counterexamples.
We construct examples of number fields which are not isomorphic but for which their idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group theory provided by Leonard Scott of a finite group $G$ and subgroups $H_1$ and $H_2$ which are not conjugate in $G$ but for which the $G$-module ${\mathbb Z}[G/H_1]$ is isomorphic to ${\mathbb Z}[G/H_2]$.
Let $E/\mathbb Q$ be an elliptic curve, and denote by $N(p)$ the number of $\mathbb{F}_p$-points of the reduction modulo $p$ of $E$. A conjecture of Koblitz, refined by Zywina, states that the number of primes $p \leq X$ at which $N(p)$ is also prime is asymptotic to $C_E \cdot X / \log(X)^2$, where $C_E$ is an arithmetically-defined non-negative constant. Following Miri-Murty (2001) and others, Y.R. Liu (2006) and David-Wu (2012) study the number of prime factors of $N(p)$. We generalize their arguments to abelian varieties $A / \mathbb Q$ whose adelic Galois representation has open image in …
A Computational Framework for the Moduli of Supersingular Abelian Varieties
建立极化旗型商的结构定理，证明顶层精细正规形和拟极化的可计算下降准则，完成旗型第一和最后一步分类，并计算 $g\leq 5$ 维的极化旗型商。
On a refinement of the Birch and Swinnerton-Dyer Conjecture in positive characteristic
本文提出正特征函数域上阿贝尔簇的 Birch 和 Swinnerton-Dyer 猜想精细化版本，融合 $L$-级数首项同余与 Selmer 复形 Galois 结构限制，并在假定 Tate-Shafarevich 群有限的条件下对一类重要例子给出完整证明。
We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult archimedean considerations related to refinements of the conjecture of De…
Evaluating the wild Brauer group
本文研究 p-adic 域上簇的 Brauer 群滤过，将其与 Kato 的 Swan 导子联系起来，并给出精细 Swan 导子的几何刻画及数域有理点应用。
A celebrated theorem of Bogomolov asserts that the $\ell$-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic $p$: a "counterexample" is provided by an ordinary elliptic curve defined over a finite field. In this note we discuss (and explicitly construct) more interesting examples of "non-constant" absolutely simple abelian varieties (without homotheties) over global fields in characteristic $p$.
Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in $\mathbb{Z}_p$
本文针对Liu和Qian关于$\mathbb{Z}_p$中受限和集临界对的猜想，在边界情形给出反例，并在$|A|+|B|\le p-1$假设下证明改进的逆定理。
We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\barρ$ of a $p$-adic field $K$ and a cocharacter $μ$. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied with $n=3$ or $μ$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\barρ$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.
We review, under a perspective which appears different from previous ones, the so-called Hilbert Property (HP) for an algebraic variety (over a number field); this is linked to Hilbert's Irreducibility Theorem and has important implications, for instance towards the Inverse Galois Problem. We shall observe that the HP is in a sense `opposite' to the Chevalley-Weil Theorem, which concerns unramified covers; this link shall immediately entail the result that the HP can possibly hold only for simply connected varieties (in the appropriate sense). In turn, this leads to new counterexamples to the …
Geometrically simple counterexamples to a local-global principle for quadratic twists
针对二次扭局部-整体原理的反例非几何单的问题，研究者利用伽罗瓦上同调与类域论，构造出满足$p\equiv13\pmod{24}$的素数$p$对应的维数为$p-1$的$\mathbb{Q}$上几何单阿贝尔簇反例对。
Existence of global Néron models beyond semi-abelian varieties
本文证明了完美剩余域情形下非半阿贝尔代数群的Bosch-Lütkebohmert-Raynaud猜想，给出了非完美剩余域的反例，并分类了完美剩余域上的单有理有界幂幺群。
On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties
本文推翻了关于周期点高度有界的猜想，并证明了在射影簇上某些主导有理自映射的周期点集高度有界，同时指出前周期点可能不满足这一性质。
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