arXiv subjects
Publications and source records attributed to Congling Qiu.
Over an infinite field, we prove a refinement of Matsusaka's theorem in which the complementary isogeny factor can be made absolutely simple of arbitrarily large dimension. We also discuss related questions around relative simplicity and the isogeny Schottky problem.
In this paper, we obtain an explicit formula for the theta correspondence of unipotent principal-series representations between an even orthogonal and a symplectic group or between general linear groups over a finite field. The formula is in terms of the Springer correspondence. Along the way we prove general results about module categories of Hecke categories arising from spherical varieties, and give a similar formula for the multiplicities of the unipotent principal series representations in the function space of the spherical variety in terms of relative Springer theory.
We use hyperelliptic Shimura curves to find triple product $L$-functions of Hilbert newforms with central vanishing orders proved to be at least 3.
A result of Green and Griffiths states that for the generic curve $C$ over $\mathbb{C}$ of genus $g \geq 4$ with a canonical divisor $K$, its Faber--Pandharipande 0-cycle $K\times K-(2g-2)K_\Delta$ on $C\times C$ is nontorsion in the Chow group of rational equivalence classes. However, according to a conjecture of Beilinson and Bloch, this Chow cycle vanishes if the curve is defined over a number field. We give a proof of this prediction for Shimura curves which have real multiplication. Our method also works for some other classes curves with partial real multiplication. We also draw a connection between the Faber--Pandharipande 0-cycles and torsion points on curves under the Abel--Jacobi map.
Associated to an algebraic curve $X$, there are two canonically constructed homologically trivial algebraic $1$-cycles, the Ceresa cycle in the Jacobian of $X$, and the Gross-Kudla-Schoen modified diagonal cycle in the triple product $X \times X \times X$. By a result of Shou-Wu Zhang, one is torsion if and only if the other is. In this paper, we prove that these two cycles associated to a large family of modular curves are non-torsion in the corresponding Chow groups. We obtain the result by relating this problem to the study of special cycles on orthogonal Shimura varieties. As the main ingredient and a result of independent interest, we develop a pullback formula for special divisors on modular curves embedded in their products via the diagonal map.
We prove that only finitely many Shimura curves can have gonality bounded by a given number, and we study the computability of this finite set. Motivated by the relation between hyperellipticity (that is, gonality 2) and the vanishing of the modified diagonal cycle, we conjecture that such vanishing occurs for only finitely many Shimura curves. We establish several finiteness and classification results toward this conjecture and, as a by-product, obtain explicit examples of curves with vanishing modified diagonal cycles. Our computations are based on modular form data from the database \textsf{LMFDB}, and some of them are carried out using the computer algebra system \textsf{Sage}.
We prove vanishing results for the modified diagonal cycles in the Chow groups of the triple products of Shimura curves and their motivic direct summands. In particular we find examples of curves with trivial automorphism groups and vanishing modified diagonal cycles.
We prove a sufficient condition for the vanishing of the modified diagonal cycle in the Chow group (with $\mathbb{Q}$-coefficients) of the triple product of a curve over $\mathbb{C}$. We exhibit infinitely many non-hyperelliptic curves, including the Fricke--Macbeath curve, the Bring curve, and two 1-dimensional families parameterized by certain Hurwitz spaces, for which our condition is satisfied.
The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of Andr\'e--Oort and Andr\'e--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.
We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at $1$ can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we generalize the results of Aubert-Michel-Rouquier and Pan on theta correspondence between the Harish-Chandra series.
We construct explicit generating series of arithmetic extensions of Kudla's special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the arithmetic Chow groups. This provides a partial solution to Kudla's modularity problem. The main ingredient in our construction is S.~Zhang's theory of admissible arithmetic divisors. The main ingredient in the proof is an arithmetic mixed Siegel-Weil formula.
We prove a higher weight general Gross--Zagier formula for CM cycles on Kuga--Sato varieties over modular curves of arbitrary levels. To formulate and prove this result, we prove several results on the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple modules whose irreducible components are associated to higher weight holomorphic cuspidal automorphic representations. These two types of results provide evidence toward two conjectures of Beilinson--Bloch. The higher weight general Gross--Zagier formula is proved using arithmetic relative trace formulas. The proof of the modularity of CM cycles is inspired by arithmetic theta lifting.
We use perfectoid spaces associated to abelian varieties and Siegel moduli spaces to study torsion points and ordinary CM points. We reprove the Manin-Mumford conjecture i.e. Raynaud's theorem. We also prove the Tate-Voloch conjecture for a product of Siegel moduli spaces namely ordinary CM points outside a closed subvariety can not be p-adically too close to it.
We prove the Gross-Zagier-Zhang formula over global function fields of arbitrary characteristics. It is an explicit formula which relates the Neron-Tate heights of CM points on abelian varieties and central derivatives of associated quadratic base change $L$-functions. Our proof is based on an arithmetic variant of a relative trace identity of Jacquet. This approach is proposed by W. Zhang.