Integrality properties in the Moduli Space of Elliptic Curves: Isogeny Case

Integrality of the Betti moduli space

If in a given rank $r$, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number $\ell$, there is an absolutely irreducible $\ell$-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank $r$ if once there is an irreducible rank $r$ complex linear representation, then for any $\ell$, there is an absolutely irre…

Families of explicitly isogenous Jacobians of variable-separated curves

We construct six infinite series of families of pairs of curves (X,Y) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3, or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou--Noguès and Couveignes' explicit classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is reducible.

Rationality and p-adic properties of reduced forms of half-integral weight

In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke operators, we derive relations of Fourier coefficients of each basis element and obtain congruences of the Fourier coefficients, which extend known congruences for traces of singular moduli.

粗 Hilbert 模概形上的整点

Integral points on coarse Hilbert moduli schemes

本文结合 Faltings 方法、模性结果与 Masser–Wüstholz 同源估计，研究粗 Hilbert 模概形分支轨迹外整点的高度与数量，并建立相关有效 Shafarevich 结果。

Explicit coverings of families of elliptic surfaces by squares of curves

We show that, for each $n>0$, there is a family of elliptic surfaces which are covered by the square of a curve of genus $2n+1$, and whose Hodge structures have an action by ${\mathbb Q}(\sqrt{-n})$. By considering the case $n=3$, we show that one particular family of K3 surfaces are covered by the square of genus $7$. Using this, we construct a correspondence between the square of a curve of genus $7$ and a general K3 surface in ${\mathbb P}^4$ with $15$ ordinary double points up to isogeny. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K…

有理点与有理模空间

Rational points and rational moduli spaces

该研究利用阿贝尔簇模空间提出代数簇的几何非退化准则，构造对应开稠模空间与有效Parsin构造，结合Faltings方法等得到有理点的高度与数量界，对大类有理曲线建立有效Mordell猜想并解决广义费马问题的ico类似问题。

The $δ$-invariant theory of Hecke correspondences on $\mathcal A_g$

Let $p$ be a prime, let $N\geq 3$ be an integer prime to $p$, let $R$ be the ring of $p$-typical Witt vectors with coefficients in an algebraic closure of $\mathbb F_p$, and consider the correspondence $\mathcal A'_{g,1,N,R}\rightrightarrows \mathcal A_{g,1,N,R}$ obtained by taking the union of all prime to $p$ Hecke correspondences on Mumford's moduli scheme of principally polarized abelian schemes of relative dimension $g$ endowed with symplectic similitude level-$N$ structure over $R$-schemes. It is well-known that the coequalizer $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ of the above co…

Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\m…

配分函数与椭圆曲线

The partition function and elliptic curves

研究通过CM点几何解释布鲁尼尔 - 奥诺公式中配分函数的迹，得到超奇异迹公式。对特殊素数\(\ell = 5,7,11\)为拉马努金分拆同余式带来新见解，提供模理论框架，核心代数恒等式已形式化验证。