Differential modular forms,Elliptic curves and Ramanujan foliation

On elliptic modular foliations

In this article we consider the three parameter family of elliptic curves $E_t: y^2-4(x-t_1)^3+t_2(x-t_1)+t_3=0, t\in\C^3$ and study the modular holomorphic foliation $\F_ω$ in $\C^3$ whose leaves are constant locus of the integration of a 1-form $ω$ over topological cycles of $E_t$. Using the Gauss-Manin connection of the family $E_t$, we show that $\F_ω$ is an algebraic foliation. In the case $ω=\frac{xdx}{y}$, we prove that a transcendent leaf of $\F_ω$ contains at most one point with algebraic coordinates and the leaves of $\F_ω$ corresponding to the zeros of integrals, never cross such a …

On elliptic modular foliations, II

We give an example of a one dimensional foliation $\cal F$ of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves $X_0(d),\ d\in\Bbb B$ minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief has been that such formulas do not exist and the emphasis in the literature has been on…

Dedekind $η$-function, Hauptmodul and invariant theory

We solve a long-standing open problem with its own long history dating back to the celebrated works of Klein and Ramanujan. This problem concerns the invariant decomposition formulas of the Hauptmodul for $Γ_0(p)$ under the action of finite simple groups $PSL(2, p)$ with $p=5, 7, 13$. The cases of $p=5$ and $7$ were solved by Klein and Ramanujan. Little was known about this problem for $p=13$. Using our invariant theory for $PSL(2, 13)$, we solve this problem. This leads to a new expression of the classical elliptic modular function of Klein: $j$-function in terms of theta constants associated…

Ramanujan向量场

Ramanujan vector field

本文证明Ramanujan向量场的不变代数曲线，并利用Cartier算子给出模空间解释，旨在推广至Calabi-Yau模形式整性猜想，提出用模向量场和Gauss-Manin联络的新证明方案。

Higher Ramanujan equations and periods of abelian varieties

We describe higher dimensional generalizations of Ramanujan's classical differential relations satisfied by the Eisenstein series $E_2$, $E_4$, $E_6$. Such "higher Ramanujan equations" are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford's theory of degenerating families of abelian varieties, we construct remarkable solutions of these dif…

关于模p的拉马努金向量场

On the Ramanujan Vector Field modulo $p$

本文研究了模p下拉马努金向量场的p次幂，并验证了一个关于该向量场的猜想，同时描述了奇数椭圆曲线的位移并证明了向量场的相切性。

Computing the Satake p-parameters of Siegel modular forms

Hecke eigenvalues of classical modular forms often encode a wealth of arithmetic data. The Satake $p$-parameters of a Siegel modular form play a role analogous to the one played by Hecke eigenvalues in the characterization of classical modular forms. In this paper we present an algorithm by which we can compute the Satake $p$-parameters of a Siegel modular form $F$ if we are given the Hecke eigenvalues of $F$ with respect to the generators of the local Hecke algebra. We compute explicit examples and relate our computations to the Ramanujan-Petersson Conjecture.

Curves in Hilbert modular varieties

We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using holomorphic foliations theory, we establish a Second Main Theorem following Nevanlinna theory. Finally, with a metric approach, we establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions.

Modular-type functions attached to mirror quintic Calabi-Yau varieties

In this article we study a differential algebra of modular-type functions attached to the periods of a one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realizati…