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# 数学代写|黎曼曲面代写Riemann surface代考|MAST90056

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## 数学代写|黎曼曲面代写Riemann surface代考|Definition and Simplest Properties

Consider a $g$-dimensional complex torus $\mathbb{C}^g / \Lambda$ where $\Lambda$ is a lattice of full rank:
$$\Lambda=A N+B M, \quad A, B \in g l(g, \mathbb{C}), N, M \in \mathbb{Z}^g,$$
and the $2 g$ columns of $A, B$ are $\mathbb{R}$-linearly independent. Non-constant meromorphic functions on $\mathbb{C}^g / \Lambda$ exist only (see, for example, [Sie71]) if the complex torus is an Abelian torus, i.e., if by an appropriate linear choice of coordinates on $\mathbb{C}^g$ the lattice (1.80) can be reduced to a special form: $A$ is a diagonal matrix of the form
$$A=2 \pi \mathrm{i} \operatorname{diag}\left(a_1=1, \ldots, a_g\right), \quad a_k \in \mathbb{N}, a_k \text { divides } a_{k+1},$$
and $B$ is a symmetric matrix with negative real part. An Abelian torus with $a_1=\ldots=a_g=1$ is called principally polarized. Jacobi varieties of Riemann surfaces are principally polarized Abelian tori. Meromorphic functions on Abelian tori are constructed in terms of theta functions, which are defined by their Fourier series.

Definition 31. Let $B$ be a symmetric $g \times g$ matrix with negative real part. The theta function is defined by the following series
$$\theta(z)=\sum_{m \in \mathbb{Z}^g} \exp \left{\frac{1}{2}(B m, m)+(z, m)\right}, \quad z \in \mathbb{C} .$$
Here
$$(B m, m)=\sum_{i j} B_{i j} m_i m_j, \quad(z, m)=\sum_j z_j m_j .$$

## 数学代写|黎曼曲面代写Riemann surface代考|Theta Functions of Riemann Surfaces

From now on we consider an Abelian torus which is a Jacobi variety, $\mathbb{C} / \Lambda=$ $J a c(\mathcal{R})$. By combining the theta function with the Abel map, one obtains the following useful map on a Riemann surface:
$$\Theta(P):=\theta\left(\mathcal{A}{P_0}(P)-d\right), \quad \mathcal{A}{P_0}(P)=\int_{P_0}^P \omega .$$
Here we incorporated the base point $P_0 \in \mathcal{R}$ in the notation of the Abel map, and the parameter $d \in \mathbb{C}^g$ is arbitrary. The periodicity properties of the theta function (1.81) imply the following

Proposition 8. $\Theta(P)$ is an entire function on the universal covering $\tilde{\mathcal{R}}$ of $\mathcal{R}$. Under analytical continuation $\mathcal{M}{a_k}, \mathcal{M}{b_k}$ along a-and b-cycles on the Riemann surface, it is transformed as follows:
$$\begin{gathered} \mathcal{M}{a_k} \Theta(P)=\Theta(P), \ \mathcal{M}{b_k} \Theta(P)=\exp \left{-\frac{1}{2} B_{k k}-\int_{P_0}^P \omega_k+d_k\right} \Theta(P) . \end{gathered}$$
The zero divisor $(\Theta)$ of $\Theta(P)$ on $\mathcal{R}$ is well defined.
Theorem 22. The theta function $\Theta(P)$ either vanishes identically on $\mathcal{R}$ or has exactly $g$ zeros (counting multiplicities):
$$\operatorname{deg}(\Theta)=g .$$
Suppose $\Theta \not \equiv 0$. As in Sect. 1.4 consider the simply connected model $F_g$ of the Riemann surface. The differential $\mathrm{d} \log \Theta$ is well defined on $F_g$ and the number of zeros of $\Theta$ is equal to
$$\operatorname{deg}(\Theta)=\frac{1}{2 \pi \mathrm{i}} \int_{\partial F_g} \mathrm{~d} \log \Theta(P) .$$

## 数学代写|黎曼曲面代写Riemann surface代考|Definition and Simplest Properties

$$\Lambda=A N+B M, \quad A, B \in g l(g, \mathbb{C}), N, M \in \mathbb{Z}^g,$$

$$A=2 \pi \mathrm{i} \operatorname{diag}\left(a_1=1, \ldots, a_g\right), \quad a_k \in \mathbb{N}, a_k \text { divides } a_{k+1},$$
$B$是一个实部为负的对称矩阵。带$a_1=\ldots=a_g=1$的阿贝尔环面称为主极化环面。黎曼曲面的雅可比变体主要是极化的阿贝尔环面。阿贝尔环面上的亚纯函数是由函数构成的，而函数是由它们的傅立叶级数定义的。

$$\theta(z)=\sum_{m \in \mathbb{Z}^g} \exp \left{\frac{1}{2}(B m, m)+(z, m)\right}, \quad z \in \mathbb{C} .$$

$$(B m, m)=\sum_{i j} B_{i j} m_i m_j, \quad(z, m)=\sum_j z_j m_j .$$

## MATLAB代写

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