Ramesh Sreekantan
We compute relations of rational equivalence among special codimensional two cycles on families of abelian surfaces using elements of higher chow groups.
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Given a newform f, we extend Howard's results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovář's extended Selmer groups attached to suitable bi…
Iwasawa theory of Heegner points on abelian varieties of GL_2 type has been studied by, among others, Mazur, Perrin-Riou, Bertolini and Howard. The purpose of this paper, the first in a series of two, is to describe extensions of some of their results in which abelian varieties are replaced by the Galois cohomology of Deligne's p-adic representation attached to a modular form f of even weight >2. In this more general setting, the role of Heegner points is played by higher-dimensional Heegner cycles in the sense of Nekovář. In particular, we prove that the Pontryagin dual of a certain Bloch-Kat…
We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated to a Shimura curve and the second term in the Laurent expansion at s=1/2 of an Eisenstein series of weight 3/2 for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions.…
In this note, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's Specialization Theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over ${\Q}$.
Non-liftability of Families of Abelian Varieties with Small $l$-adic Local System
本文研究特征 $p$ 上光滑正则曲线上的阿贝尔簇族，证明其霍奇包非 nef 且无法提升到 $W_2(k)$，并建立一个阿拉凯洛夫型不等式。
In 1983 Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has `small height' is finite. We conjecture a generalisation to higher-dimensional families, where we replace `finite' by `not Zariski dense'. We show that this conjecture would imply the Uniform Boundedness Conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.
The monodromy of families of subvarieties on abelian varieties
本文结合几何论证与表示论，证明阿贝尔簇中高余维非平凡子簇族经一般秩一局部系统扭转后具有大单值性。
Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation
本研究借助p进霍奇理论与朗兰兹对应，构造了4穿孔射影直线上无穷多族带特定坏约化的$\text{GL}_2$型阿贝尔簇，结合相关猜想证明其可由椭圆曲线泛族挠截面参数化，且可导出Painleve VI方程的代数解。
Let Z be a subvariety of the moduli space of principally polarised abelian varieties of dimension g over the complex numbers. Suppose that Z contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny class. A generalisation of a conjecture of André and Pink predicts that Z is a weakly special subvariety. We prove this when dim Z = 1 using the Pila--Zannier method and the Masser--Wüstholz isogeny theorem. This generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM points and of Pink when it consists of Galois generic points.
We consider the following question : given a family over abelian varieties $\mathcal{A}$ over a curve $B$ defined over a number field $k$, how does the rank of the Mordell-Weil group of the fibres $\mathcal{A}_t(k)$ vary? A specialisation theorem of Silverman guarantees that, for almost all $t$ in $C(k)$, the rank of the fibre is at least the generic rank, that is the rank of $\mathcal{A}(k(B))$. When the base curve $B$ is rational, we show, at least in many cases and under some geometric conditions, that there are infinitely many fibres for which the rank is larger than the generic rank. This…
Isogeny graphs associated to Moret-Bailly families of supersingular abelian surfaces
研究特征p下阿贝尔曲面的Moret-Bailly族等价类与相对(ℓ,ℓ)-同源构成的有限有向图，将其与GSp₄/ℚ的内形式上的代数模形式空间关联，并利用Jacquet-Langlands对应估计邻接矩阵的特征值。
Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we prove that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we use curves on toric surfaces to prove that liminf_{g-->infty}…
We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of projective smooth families of curves over Abelian varieties. We also study the isotriviality over simply connected projective smooth varieties.
Singular intersections in families of abelian varieties
本文证明阿贝尔概形中曲线与适当平坦子群概形的切向交点有限，并给出椭圆曲线情形的 Pila-Zannier 方法新证明，属于非预期交点问题及相对 Pink 猜想的变体。
We prove Chern class equalities for abelian families with a holomorphic normal projective connection.
$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields
本文给出了非阿贝尔三次和四次域上椭圆曲线族的两种构造，在驯顺符号核中分别获得三个和四个线性无关的“积分”元素，证明了不忽略挠元的新积分性判据，并对90余条曲线数值验证了 Beilinson 猜想。
Ranks of abelian varieties in cyclotomic twist families
该研究在伽罗瓦作用可约假设下，给出分圆扭族中阿贝尔簇Mordell--Weil群平均秩的界，结合一致Mordell--Lang猜想得到相关曲线扭族有理点个数的近一致界，核心技术是确定3-同源Selmer群的平均大小。
Finite translation orbits on double families of abelian varieties (with an appendix by E. Amerik)
本研究针对两类由公共簇上不同有理映射诱导的g维阿贝尔簇族，分析非挠截面诱导的平移群特定子集的作用，在基簇维数不超过g的条件下证明有限轨道点含于真Zariski闭子集并给出部分显式刻画，将已有定理推广至高维。
Equidistribution in Families of Abelian Varieties and Uniformity
本文利用 Arakelov 理论等分布技术推导出 Manin-Mumford 和 Bogomolov 猜想的一致性结果，并证明代数曲线上挠点和有理点数量可仅由亏格与 Mordell-Weil 秩一致界定。
In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation havi…
Given a fixed product of non-isogenous abelian varieties at least one of which is general, we show how to construct complete families of indecomposable abelian varieties whose very general fiber is isogenous to the given product and whose connected monodromy group is a product of symplectic groups or is a unitary group. As a consequence, we show how to realize any product of symplectic groups of total rank $g$ as the connected monodromy group of a complete family of $g'$-dimensional abelian varieties for any $g'\ge g$. These methods also yield a construction of a new Kodaira fibration with fib…
In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight $-1$.
Under the condition that the Prym map is injective in characteristic $p$, we prove that the special subvarieties in the moduli space of abelian varieties of dimension $l$ and polarization type $D$, $A_{l,D}$, arising from families of abelian covers of $¶^1$ are of a very restrictive nature. In other words, if the family is one-dimensional or if it contains an eigenspace of certain type for the group action on the cohomology of fibers, then the Shimura varieties arising from such families can only be constructed by the group action of the family.
Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures
本文在奇特征整体函数域上研究几乎普通阿贝尔曲面，证明在无穷多个位处其模约化具有包含Z[x]/(x^2-m)的自同态环从而非单，并将椭圆曲线的S-整性定理推广到阿贝尔曲面。
Finite monodromy groups in degenerating families and the stability degree of curves
该研究计算亏格g的曲线与指定秩的半阿贝尔簇的稳定度，证明前者等于Minkowski界，通过退化族有限单值群的 specialization 结果结合素数2处显式构造完成证明。
We describe an algorithm for computing the Picard-Fuchs equation for a family of twists of a fixed elliptic surface. We then apply this algorithm to obtain the equation for several examples, which are coming from families of Kummer surfaces over Shimura curves, as studied in our previous work. We use this to find correspondenced between the parameter spaces of our families and Shimura curves. These correspondences can sometimes be proved rigorously.
Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset U of S to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne-Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily-Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.
We prove an absolute Chow-Kuenneth decomposition for the motive of universal families A of abelian varieties over some compact Shimura surface. We furthermore prove the Hodge conjecture for general fibres of A, extending results of Ribet.
In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on …
Let $E_λ$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-λ)$. We recently proved that, given $n$ linearly independent points $P_1(λ), \dots,P_n(λ)$ on $E_λ$ with coordinates in $\bar{\mathbb{Q}(λ)}$, there are at most finitely many complex numbers $λ_0$ such that the points $P_1(λ_0), \dots,P_n(λ_0)$ satisfy two independent relations on $E_{λ_0}$. In this article we continue our investigations on Unlikely Intersections in families of abelian varieties and consider the case of a curve in a product of two non-isogenous families of elliptic curves and in a family of split semi-abelian v…
We study the locus of abelian Galois covers of $\mathbb{P}^{1}$ in $A_{g}$ and the problem of occurrence of Shimura (special) subvarieties generated by these covers in the Torelli locus $T_{g}$ inside $A_{g}$. We first investigate the existence of Shimura subvarieties in the mentioned locus by some computational methods based on Moonen-Oort works and then exclude many cases using both characteristic $p$ methods and monodromy computations.
This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct images of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of …
We show that Ribet sections are the only obstruction to the validity of the relative Manin-Mumford conjecture for one dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber-Pink conjecture for curves in a mixed Shimura variety of dimension four, as well as the study of polynomial Pell equations with non-separable discriminants.
Let $A$ be abelian variety over the function field $K$ of a compact Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and a certain effective horizontal divisor $\DD \subset \mathcal{A}$. We give a sufficient condition on the divisor $\DD$ for the finiteness of the set of $(S, \DD)$-integral sections for every finite subset $S \subset B$. These integral sections $σ$ correspond to rational points in $A(K)$ which satisfy the geometric condition $f ( σ(B) \cap \DD)\subset S$. This notion is the geometric variant of integral solutions of a system of \emph{Diophantine equations…
Atkin polynomials for families of abelian varieties with real multiplication
本文通过正交多项式与帕德逼近的关系，研究了具有实乘法的阿贝尔簇族的非普通域几何，并应用超几何恒等式确定了阿特金多项式的递推关系。
The Poincaré torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the group subschemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink's conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformization of the Poincaré torsor, but other, more geometric, approache…
Families of automorphisms of abelian varieties
本文研究复单位圆盘参数化的极化阿贝尔簇族上自同构动力学的代数性质，证明无分圆因子时映射可正则化当且仅当族不退化，并证明平移族总可正则化，同时刻画了轨道闭包。
We extend the Serre-Tate theory of canonical lifts of ordinary abelian varieties to arbitrary unpolarised families of ordinary abelian varieties parameterised by a $p$-adic formal scheme $S$. We show that the canonical lift is the unique lift to $W(S)$ which admits a $δ$-structure in the sense of Joyal, Buium, and Bousfield. We prove analogous statements for families of ordinary $p$-groups and $p$-divisible groups.
We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman- Vladuts theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P.Serre, using results of R.M.Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for…
Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the Arakelov inequalities say that 2deg(F(1,0)) is bounded from above by g=rank(F(1,0))(2q-2+s). We show that for s>0 families reaching this bound are isogenous to the g-fold product of a modular family of elliptic curves, and a constant abelian variety. The content of this note became part of the article "A characterization of certain Shimura curves in the moduly sta…
We prove that under some conditions on the monodromy, families of abelian covers of the projective line do not give rise to (higher dimensional) Shimura subvarieties in $A_g$. This is achieved by a reduction to $p$ argument. We also use another method based on monodromy computations to show that two dimensional subvarieties in the above locus are not special. In particular it is shown that such families have usually large monodromy groups. Together with our earlier results, the above mentioned results contribute to classifying the special families in the moduli space of abelian varieties and p…
We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behaviour of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which resu…
We extend the Spectral Construction, a technique used with great success to study and construct vector bundles on elliptically fibered varieties, to a special family of abelian surface fibered varieties. The results are motivated by requirements from Heterotic String Phenomenology where vector bundles with specified chern class are required to produce a realistic particle spectrum. Although only certain (1,8)-polarized families of abelian surfaces are considered here we expect the main ideas to carry over to other families of abelian varieties with a uniform relative polarization type.
So far only six families of smooth irregular surfaces are known to exist in P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10), two different families of non-minimal abelian surfaces of degree 15, and one family of non-minimal bielliptic surfaces of degree 15. The main purpose of the paper is to describe the structure of the Hartshorne-Rao modules and the syzygies for each of these smooth irregular surfaces in P^4, providing at the same time a unified construction method (via syzygies) for t…
Gromov-Witten theory of abelian varieties in families and modular forms
本文为系列研究首篇，将椭圆曲线格罗莫夫-维滕理论拟模性推广至高维，经重言投影后证明亏格2情形的相关猜想，引入商格罗莫夫-维滕不变量并给出其显式公式。
Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields
研究函数域上阿贝尔簇二次扭转族塞尔默群，证明Bhargava-Kane-Lenstra-Poonen-Rains启发法版本，得出关于塞尔默秩“极简主义猜想”的结果，关键输入是广义赫维茨空间同调稳定性定理及塞尔默群平均大小表达式。
In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic fourfolds. This also implies abelianity and finite dimensionality of the motive of related Hyperkähler varieties, such as the Fano variety of lines and the LLSvS 8fold.
In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.
K3 surfaces and cubic fourfolds with Abelian motive
本文证明高代数2-圈秩的三次四维流形及其相关超凯勒簇具有阿贝尔且有限维的Chow动机，并从四维K3曲面族构造出两族动机有限维的Fano簇。
We consider families of cyclic covers of the projective line, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques in mixed characteristics due to Dwork and Ogus.
We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by Z[\sqrt{2}].
We discuss the existence of an absolute Chow-Kuenneth decomposition for complete degenerations of families of Abelian threefolds with complex multiplication over a particular Picard Modular Surface studied by Holzapfel. In addition to the work of Gordon, Hanamura and Murre we use Relatively Complete Models in the sense of Mumford-Faltings-Chai of Picard Modular Surfaces in order to describe complete degenerations of families of abelian varieties. We furthermore prove vanishing results for cohomology groups of irreducible representations of certain arithmetic subgroups in SU(2,1) using the non-…
Using abelian differentials and periods of the universal Mumford curve, we study the universal expression and asymptotic behavior of tau functions defined for stably degenerating families of algebraic curves with additional data. Furthermore, we apply this study to constructing solutions to the KP hierarchy which are expressed by nonarchimedean theta functions and by mixtures of quasi-periodic solutions with solitons containing real solutions defined on families of M-curves.
Suppose that g > 2, that n > 0 and that m > 0. In this paper we show that if E is an irreducible smooth variety which dominates a divisor D in M_{g,n}[m], the moduli space of n-pointed, smooth curves of genus g with a level m structure, then the closure of the image of the monodromy representation pi_1(E,e)--> Sp_g(Zhat) has finite index in Sp_g(Zhat). A similar result is proved for codimension 1 families of the universal principally polarized abelian variety of dimension g > 2. Both results are deduced from a general "non-abelian strictness theorem". The first result is used in arXiv:1001.500…
On Shimura curves generated by families of Galois $G$-covers of curves
本文利用 Higgs 丛理论和 Viehweg-Zuo 刻画，证明了在特定素因子条件下，椭圆曲线及任意亏格曲线的 $\Z_n$-覆盖族无法生成位于 Torelli 轨迹中的非紧 Shimura 曲线。
We construct families of principally polarized abelian varieties whose theta divisor is irreducible and contains an abelian subvariety. These families are used to construct examples when the Gauss map of the theta divisor is only generically finite and not finite. That is, the Gauss map in these cases has at least one positive-dimensional fiber. We also obtain lower-bounds on the dimension of Andreotti-Mayer loci.
Abelian varieties and Riemann surfaces with generalized quaternion group action
本文研究广义四元数群作用下的 Riemann 曲面与 Abel 簇，给出同源分解、Jacobian 与零亏格作用的联系、复一维族的完整分类，并构造显式族及亏格四超椭圆曲面 Jacobian 的周期矩阵。
We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus $σ\geq 2$ which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.
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