Related papers: On a simple model of X_0(N)

In this paper, we define the normalized Eisenstein series $\mathcal{P}$, $e$, and $\mathcal{Q}$ associated with $\Gamma_0(2),$ and derive three differential equations satisfied by them from some trigonometric identities. By using these…

We prove in one go that each of the 4 families of Burniat surfaces with K^2 = 6,5,4, is a connected component of the moduli space of surfaces of general type. We prove also the rationality of each component. In the nodal case (one of the…

Let X be a smooth cubic threefold, M the moduli space of stable rank 2 vector bundles on X with trivial determinant and c_2=2 (the smallest value for which this space is non-empty). Recent results of Druel, Iliev, Markushevich and…

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations,…

We prove that, given integers $m\geq 3$, $r\geq 1$ and $n\geq 0$, the moduli space of torsion free sheaves on $\mathbb P^m$ with Chern character $(r,0,\ldots,0,-n)$ that are trivial along a hyperplane $D \subset \mathbb P^m$ is isomorphic…

Let (X,L) be a quasi polarized pairs, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles K_X + rL for high rational…

Let X be a smooth projective curve. Write Bun_{SO_{2n}} for the moduli stack of SO_{2n}-torsors on X. We give a geometric interpretation of the automorphic function f on Bun_{SO_{2n}} corresponding to the minimal representation. Namely, we…

Given an integer $N \geq 3$, we prove that for any ring $R$ and any finite locally free $R$-group scheme $G$ which is fppf-locally (over $R$) isomorphic the $N$-torsion subscheme of some elliptic curve $E/R$, there is a smooth affine curve…

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify…

In this paper we consider the birational classification of pairs (S,L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its…

Let $M_1$ be the moduli space of the KSBA stable surfaces $X$ of geometric genus $p_g(X)=1$ realizing the minimal possible volume $K_X^2=\frac1{143}$. We show that its reduced part $M_{1,\rm red}$ is a $10$-dimensional projective variety…

We establish two families of congruences modulo powers of 5 for the Fourier coefficients of $(2E_2(2\tau)-E_2(\tau))\eta(2\tau)^{-1}$, where $E_2(\tau)$ is the weight 2 Eisenstein series and $\eta(\tau)$ is the Dedekind eta function. This…

We consider minimal maps $f:M\to N$ between Riemannian manifolds $(M,\mathrm{g}_M)$ and $(N,\mathrm{g}_N)$, where $M$ is compact and where the sectional curvatures satisfy $\sec_N\le \sigma\le \sec_M$ for some $\sigma>0$. Under certain…

In the classical setting, the modular equation of level $N$ for the modular curve $X_0(1)$ is the polynomial relation satisfied by $j(\tau)$ and $j(N\tau)$, where $j(\tau)$ is the standard elliptic $j$-function. In this paper, we will…

We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth…

Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex…

The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$ there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is the conductor of $E$. These maps may be expressed in terms of pairs of modular…

Let $\mathcal{X}_0(N)$ be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational $N$-isogeny over a number field $F$. Let $N\in\{1,2,3,4,5,6,7,8,9,10,12,13,16,18,25\},$ which are precisely the values for which…

The orbifold generalization of the partition function, which would describe the gauge theory on the ALE space, is investigated from the combinatorial perspective. It is shown that the root of unity limit of the q-deformed partition function…

We describe a higher dimensional generalization of Ramanujan's differential equations satisfied by the Eisenstein series $E_2$, $E_4$, and $E_6$. This will be obtained geometrically as follows. For every integer $g\ge 1$, we construct a…