Andrei Yafaev
In this paper we prove, assuming the Generalised Riemann Hypothesis, a conjecture of Yves Andre that that asserts that a curve in a Shimura variety containing an infinite set of special points is of Hodge type.
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The fullness conjectures for products of elliptic curves
本文证明了伊夫·安德烈 motive 著作第7章中所有猜想在椭圆曲线乘积情形下成立，且证明比已知情形的先前证明更简单统一。
A conjecture by Yves Andre and Frans Oort says that closed subvarieties of Shimura varieties that contain a Zariski dense subset of special points are subvarieties of Hodge type. We prove this in the case where the subvariety is a curve that contains infinitely many special points that lie in one Hecke orbit. Via work of Juergen Wolfart, Paula Cohen and Gisbert Wuestholz, this result has an application to algebraicity of values of hypergeometric functions.
We give a proof of the André-Oort conjecture for $\mathcal{A}_g$ - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged' version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The André-Oort conjecture then follows from previous work of Pila and the author.
Canonical Heights on Shimura Varieties and the André-Oort Conjecture
本文旨在完全一般性地证明 André-Oort 猜想，该猜想涉及 Shimura 簇上特殊子簇的分布。
We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a different approach. Then combining these results we prove the André-Oort conjecture for any mixed Shimura variety whose pure part is a subvariety of A_6^n.
Height functions on Hecke orbits and the generalised André-Pink-Zannier conjecture
本文在Shimura簇中定义广义Hecke轨道及其高度函数，在Mumford-Tate猜想版本下给出Galois轨道大小下界，并借Pila-Zannier策略证明广义André-Pink-Zannier猜想。
A characteristic p analogue of the André--Pink--Zannier conjecture
该研究针对特征p下的André--Pink--Zannier猜想，证明了Hodge型Shimura簇中具大单值群的常值函数域点情形及相应Hecke等分布代数类似物，通过Hecke对应分析与Weyl特殊点充足性证明完成论证。
Let $Z=X_1\times...\times X_n$ be a product of Drinfeld modular curves. We characterize those algebraic subvarieties $X \subset Z$ containing a Zariski-dense set of CM points, i.e. points corresponding to $n$-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic $p$ analogue of a special case of the André-Oort conjecture. We follow closely the approach used by Bas Edixhoven in characteristic zero, see math.NT/0302138. Note that in this paper we assume that the characteristic $p$ is odd, and we only treat the case of Drinfeld $F_q[T]$-mo…
Functional Transcendence of Periods and the Geometric André--Grothendieck Period Conjecture
本文通过证明代数簇族中代数形式积分的函数超越性，给出混合 Hodge 结构变分的 Ax–Schanuel 结果，并据此建立几何形式的 André–Grothendieck 周期猜想。
Generalised André-Pink-Zannier Conjecture for Shimura varieties of abelian type
本文证明了所有阿贝尔型 Shimura 簇的广义 André-Pink-Zannier 猜想，提出了适用于所有 Shimura 簇的通用命题，并证明了关于 p-adic 整数上齐次空间良好约化与 Mumford 稳定性关系的 p-adic Kempf-Ness 定理。
We state and prove a variant of the André-Oort conjecture for the product of 2 modular curves in positive characteristic, assuming GRH for quadratic fields.
We provide an unconditional proof of the André-Oort conjecture for the coarse moduli space $\mathcal{A}_{2,1}$ of principally polarized Abelian surfaces, following the strategy outlined by Pila-Zannier.
In this paper we prove, assuming the Generalized Riemann Hypothesis, the Andr?e-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH.
In this paper we develop a strategy and some technical tools for proving the Andre-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special subvarieties of Shimura varieties, assuming the Generalised Riemann Hypothesis. We proceed to show that sequences of special subvarieties whose Galois orbits have bounded degrees are equidistributed in a suitable sense.
In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of degree d polynomials, the subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case of cubic polynomials.
Proof of a conjecture of Andrews and Bachraoui on a Hecke sum
本文证明了安德鲁斯和巴赫拉乌伊关于两种颜色划分生成函数与Hecke型双和关系的猜想，采用Zwegers的不定theta函数理论及模假theta函数的模变换性质。
We establish the equidistribution with respect to the bifurcation measure of post-critically finite maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $M_2$ (or generalized Mandelbrot set $M_d$ for degree $d>2$), we classify all complex plane curves $C$ with Zariski-dense subsets of points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously post-critically finite for a fixed degree $d\geq 2$. Our result is …
Proof of the Andrews-El Bachraoui positivity conjecture
通过q-超几何方法证明了安德鲁斯-埃尔巴哈里oui正性猜想，证实了该猜想的正确性。
A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture
本文证明了关于双 Lambert 级数的 Amdeberhan-Andrews-Ballantine 猜想的一个推广，通过建立系数由广义除数函数 σ_k(n) 给出的恒等式，将原猜想作为特例包含在内。
We prove, assuming the generalized Riemann hypothesis, the Andre-Oort conjecture for Hilbert modular surfaces. More precisely, let K be a real quadratic field and let S be the coarse moduli space of complex abelian surfaces with multiplications by the ring of integers of K. Let C be an irreducible closed curve in S, and suppose that C contains infinitely many complex multiplication points. Then we prove, assuming GRH, that C is of Hodge type, meaning, in this case, that it parametrizes abelian varieties with more endomorphisms. Also, if we assume that C has infinitely many CM points that corre…
Proof of a conjecture of Andrews and El Bachraoui on the parity of two-color partitions
研究安德鲁斯和埃尔·巴赫拉维关于双色划分奇偶性的猜想，通过证明若\(q\)-级数傅里叶系数\(t_o(n)\)为奇数，\(8n + 9\)可由\(x^2 + 2y^2\)表示来进行验证。
We prove, following Deligne and André, that the Hodge classes on abelian varieties of CM-type can be expressed in terms of divisor classes and split Weil classes, and we describe some consequences. In particular, we show that Grothendieck's standard conjecture of Lefschetz type implies the Hodge conjecture for abelian varieties (Abdulali, André, ...).
We demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multiplier of very low degree, except those which vanish, generalizing results of Baker and DeMarco, and Favre and Gauthier.
We prove a special case of the Dynamical Andre-Oort Conjecture formulated by Baker and DeMarco. For any integer d>1, we show that for a rational plane curve C parametrized by (t, h(t)) for some non-constant polynomial h with complex coefficients, if there exist infinitely many points (a,b) on the curve C such that both z^d+a and z^d+b are postcritically finite maps, then h(z)=uz for a (d-1)-st root of unity u. As a by-product of our proof, we show that the Mandelbrot set is not the filled Julia set of any polynomial with complex coefficients.
Exceptional Sign Pairs in Oscillatory Asymptotics and Conjectures of Andrews
研究拉马努金遗失笔记本中$v_2(q)$、$v_3(q)$和$v_4(q)$这三个$q$级数的特殊符号对现象，结合渐近展开与振荡因子分析，证明特殊指标以结构化方式无限次出现，存在无穷多对同号连续系数且至少一个是局部最小值。
Two Proofs of a Conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series and a new Representation for $E_2(q)$
本文证明了Amdeberhan、Andrews和Ballantine关于双Lambert级数的一个猜想，并给出了两个证明，其中一个推导出拟模形式$E_2(q)$的新表示。
In unpublished notes, Pila discussed some theory surrounding the modular function $j$ and its derivatives. A focal point of these notes was the statement of two conjectures regarding $j$, $j'$ and $j"$: a Zilber-Pink type statement incorporating $j$, $j'$ and $j"$, which was an extension of an apparently weaker conjecture of Andre-Oort type. In this paper, I first cover some background regarding $j$, $j'$ and $j"$, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila-Zannier strategy to prove a weakened version of Pila's "Modular Andre-Oo…
Heights on 'Hybrid orbits' in Shimura varieties
本文在 abelian 型 Shimura 簇情形下证明“混合猜想”，统一推广了 Andreé-Oort 与 André-Pink-Zannier 两个猜想。
We prove the unpolarized Shafarevich conjecture for K3 surfaces: the set of isomorphism classes of K3 surfaces over a fixed number field with good reduction away from a fixed and finite set of places is finite. Our proof is based on the theorems of Faltings and André, as well as the Kuga-Satake construction.
Two $q$-congruences from Jackson's ${}_8ϕ_7$ summation and Andrews' ${}_4ϕ_3$ summation
本文证明了郭关于截断基本超几何级数的两个 $q$-同余猜想，分别将模从 $Φ_n(q)^4$ 加强到 $Φ_n(q)^5$（$n\equiv3\pmod5$）和证明了模 $Φ_n(q)$ 的循环同余式（$n\equiv1\pmod4$）。
In this paper, we prove the bounded case of the Andre-Oort conjecture for special subvarieties in a mixed Shimura variety. This generalizes previous results of L. Clozel, E. Ullmo, and A. Yafaev. The proof is reduced to a special case of the equidistribution conjecture for mixed Shimura varieties. The main tool is a theorem of S. Mozes and N. Shah for certain measures on lattice spaces.
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