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Every holomorphic map $V_1/{\Lambda }_1\to V_2/{\Lambda }_2$ lifts to $z\mapsto Az+x$, where $A$ is a complex-linear map satisfying $A{\Lambda }_1\subseteq {\Lambda }_2$. Period differences of a lift lie in the discrete target Euclidean lattice, so its derivative is periodic. Boundedness on a compact fundamental domain and the Liouville theorem make this derivative constant. The linear map is unique, and the translation vector is determined modulo ${\Lambda }_2$.

Let ${\pi }_{\Lambda }:\mathbb{C}\to \mathbb{C}/\Lambda$ and ${\pi }_{{\Lambda }^{\prime }}:\mathbb{C}\to \mathbb{C}/{\Lambda }^{\prime }$ be the universal covering maps of two one-dimensional complex tori. Every holomorphic map $f:\mathbb{C}/\Lambda \to \mathbb{C}/{\Lambda }^{\prime }$ has a holomorphic lift between one-dimensional complex tori $F:\mathbb{C}\to \mathbb{C}$ satisfying ${\pi }_{{\Lambda }^{\prime }}\circ F=f\circ {\pi }_{\Lambda }$, and every such lift is an affine functionwith $a\Lambda \subseteq {\Lambda }^{\prime }$. Indeed, for $\lambda \in \Lambda$, the continuous function $F(z+\lambda )-F(z)$ takes values in the discrete set ${\Lambda }^{\prime }$ and is therefore constant. Hence $F^{\prime }$ is $\Lambda$-periodic. It is bounded on a compact fundamental parallelogram and therefore on the complex plane, so the Liouville theorem makes $F^{\prime }$ constant. If $f$ preserves the identity and the lift is chosen with $F(0)=0$, then $b=0$ and the lift is a linear map.

For a holomorphic map between metric Riemann surfaces, the average sheet number is the pulled-back source area divided by the target area. The area formula identifies it with the average number of preimages counted with multiplicity. The same definition over a target region $D$ gives $S_D=A_f(f^{-1}D)/A(D)$.

A biholomorphism is a bijective holomorphic map whose inverse is holomorphic. A degree-one nonconstant holomorphic map between compact connected Riemann surfaces is a biholomorphism.

Suppose a nonconstant holomorphic map $f:X\to {\mathbb{P}}^1$ from a connected compact Riemann surface has degree of a holomorphic map two and exactly four distinct branch values of a holomorphic map. The valency theorem forces each of their fibres to contain one ramification point with $e_p=2$, a simple ramification point of a holomorphic map. HenceThis determines the genus without recognizing a torus in a drawing.

A concrete realization is the compact Riemann surface obtained by adding the two points at infinity to the even-degree hyperelliptic model given by the algebraic curveProjection to $z$ has degree of a holomorphic map two. At a simple zero $a$ of $P$, $w$ is a local coordinate and $z-a=w^2/P^{\prime }(a)+O(w^4)$, so $e_p=2$. At infinity set $u=1/z$ and $s=w/z^2$: then $s^2=(1-4u^2)(1-u^2)$. There are two points with $u=0$, $s=\pm 1$, each unramified since $u$ is a local coordinate. The four finite roots are therefore all the branch values of a holomorphic map.^[ref]

Choose a triangulation of a surface on the Riemann sphere with these four branch values of a holomorphic map as vertices, combinatorially the boundary of a tetrahedron. Its counts are $V=4,E=6,F=4$. Upstairs, each vertex has one preimage and each open edge and face has two lifts: $V_X=4,E_X=12,F_X=8$, giving Euler characteristic $4-12+8=0$. This is a cell schematic, not a geometric immersion of $X$. The Triangulation proof of the Riemann-Hurwitz formula applies the same counting idea after refining a general triangulation of a surface around the branch values of a holomorphic map.^[ref]

Let $f$ be a holomorphic function on $0<|z-a|<R$, and suppose $a$ is an isolated essential singularity. There is a set $E\subseteq \mathbb{C}$ with at most one element such that, for every $w\notin E$ and every $0<r<R$, the equation $f(z)=w$ has infinitely many distinct solutions with $0<|z-a|<r$. Thus each nonexceptional value has preimages accumulating at $a$ in every punctured neighbourhood. The same possible exception works for all these neighbourhoods; it need not be a completely omitted value, since it could be attained only finitely often near $a$. ^[ref]

The basic example is $f(z)=e^{1/z}$ at $a=0$. It is a holomorphic function for $z\ne 0$, and its Laurent serieshas infinitely many nonzero negative-power terms, making zero an essential singularity. Its sole omitted value is zero. For the value one, its preimages are exactlyThey satisfy $e^{1/z_k}=1$ and $|z_k|=1/(2\pi |k|)\to 0$. For any $r>0$, every integer $k$ with $|k|>1/(2\pi r)$ therefore supplies a distinct solution inside $0<|z|<r$.

The diagram displays only finitely many preimages; the formula and its limit, rather than the finite picture, establish that infinitely many occur in every punctured neighbourhood. More generally, for any $w\ne 0$, choose an argument $\theta$ of $w$. All its preimages arewith any zero denominator excluded. Such an exclusion occurs only for $w=1$, with exactly one excluded integer $k$ for the chosen argument. The remaining preimages are distinct and tend to zero as $|k|\to \infty$.

The isolated essential singularity hypothesis is crucial. The function $f(z)=z^{-1}$ has a pole at zero; on $0<|z|<r$ its image is only $\{w:|w|>1/r\}$. It omits every value in the closed disc $|w|\le 1/r$, and each attained value has just one preimage. A pole therefore has entirely different local behaviour from an essential singularity.

The Casorati-Weierstrass theorem says that the image near an essential singularity is a dense subset of the complex plane. The Great Picard theorem replaces arbitrary approximation by exact attainment, and then repeats that attainment infinitely often. This is useful when a local restriction on the image can rule out an essential singularity and reduce the possibilities to a pole or a removable singularity. For a meromorphic function viewed as a holomorphic map into the Riemann sphere, the corresponding statement has at most two exceptional sphere values, including infinity in the count; it applies when the map has an isolated essential singularity, even if poles accumulate there. Picard's 1880 memoir proves the local result in section 15, treating the cases with and without accumulating poles separately. ^[ref]

The Great Picard theorem also explains the Little Picard theorem. For a nonconstant polynomial, every value is attained by the fundamental theorem of algebra. For an entire function which is not a polynomial, the function $g(\zeta )=f(1/\zeta )$ has an essential singularity at zero: a removable singularity would force $f$ to be constant by the Liouville theorem, and a pole would force $f$ to be a polynomial. Applying the Great Picard theorem to $g$ leaves at most one finite omitted value for $f$.

A holomorphic bundle map is a holomorphic map between total spaces of holomorphic vector bundles that covers a holomorphic base map and is complex-linear on each fibre. Over a fixed base, its matrix in holomorphic local frames has holomorphic entries.

A holomorphic embedding is a holomorphic map that is also a smooth embedding. Its image is a complex submanifold and the map is a biholomorphism onto that image.

A map between complex manifolds is holomorphic when its local coordinate expression has holomorphic scalar components. For a smooth map, this is equivalent to its differential being complex-linear at every point. These maps compose by the chain rule. For one-dimensional manifolds this reduces to a holomorphic map between Riemann surfaces; the general definition also applies to multiplication and inversion on a complex Lie group.

Every holomorphic map $f:\mathbb{C}\to \mathbb{C}/\Lambda$ is constant or surjective. It lifts to an entire function $F:\mathbb{C}\to \mathbb{C}$. If $F$ is nonconstant, the Little Picard theorem says it omits at most one point, while every fiber of $\mathbb{C}\to \mathbb{C}/\Lambda$ is an infinite lattice coset; hence every point of the torus has a preimage under $f$.

On the unit sphere, use $\zeta =(x+iy)/(1-z)$ away from the north pole and $\eta =(x-iy)/(1+z)$ away from the south pole. Their overlap relation $\eta =1/\zeta$ is a holomorphic map, giving a holomorphic atlas for the Riemann sphere. The opposite sign of $iy$ in the second manifold chart ensures a holomorphic rather than antiholomorphic transition.

Writing the sphere coordinates as $(X,Y,Z)$ avoids confusing their height with the complex coordinate $z$. The two manifold charts cover the whole unit sphere:On their overlap, $X^2+Y^2=1-Z^2$ givesThe coordinate transformation and its inverse are holomorphic, so these manifold charts make the sphere a Riemann surface. The sign in $X-iY$ is essential. Geometric projection from the south pole with the same planar axis labels would give $v=(X+iY)/(1+Z)=1/\overline{z}$. The complex conjugation in that transition prevents it from being holomorphic. For $z=i$, the correct second coordinate is $w=-i$: positive $Y$ corresponds to negative imaginary part in this manifold chart.

For a nonconstant holomorphic map into the unit disc, the displayed density pulls the target hyperbolic metric back to the source Euclidean coordinate. It is not the operator norm between two hyperbolic tangent spaces, which additionally divides by the source metric density. Away from ramification points $\Delta \log H_g=H_g^2$; a ramification point of multiplicity $m$ adds $2\pi m$ times its point mass to the distributional Laplacian.

A closed analytic subset of a complex manifold is nonsingular when every point has a neighbourhood on which the subset is the zero set of a holomorphic map of full rank along that zero set. The holomorphic implicit function theorem then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.

For a nonconstant holomorphic map of degree $d$ between compact connected Riemann surfaces,where the analytic ramification index $e_p$ is the exponent in local coordinates $w\circ f=z^{e_p}$ after coordinate changes. A meromorphic function maps its poles to infinity; the target coordinate $1/w$ is then holomorphic and vanishes at a pole, so this defines a holomorphic map to the Riemann sphere. A pole of order $m$ has local degree $m$.

Let $\Lambda$ be the period lattice. The induced function $\wp :T=\mathbb{C}/\Lambda \to {\mathbb{CP}}^1$ has just one pole, of order two, so its degree is two. The Riemann-Hurwitz formula gives total ramification four. The pole contributes one; every analytic ramification index is at most two, so there are exactly three further ramification points, each simple. Evenness of $\wp$, obtained by reindexing its lattice sum, identifies them with the three nonzero half-periods. Their values $e_1,e_2,e_3$ are distinct: two points of local degree two cannot lie in one degree-two fiber.

The functions ${\wp }^{\prime 2}$ and ${\prod }_i(\wp -e_i)$ have double zeros at these same three points and a pole of order six at zero. Their ratio is a holomorphic nowhere-zero function on the compact torus, hence constant. From $\wp (z)=z^{-2}+O(z^2)$ and ${\wp }^{\prime }(z)=-2z^{-3}+O(z)$ its value is four. ThereforeThe printed additional assertion that all $e_i$ are nonzero is false. The square-lattice Weierstrass function has a zero branch value: for $\mathbb{Z}+i\mathbb{Z}$, reindexing gives $\wp (iz)=-\wp (z)$. At the half-period $h=(1+i)/2$, one has $ih=h-1$, so periodicity gives $\wp (h)=-\wp (h)$ and hence $\wp (h)=0$. Thus one branch value is zero, while the three remain distinct.

The derivative has a single triple pole, so $\mathrm{deg}{\wp }^{\prime }=3$, and the Riemann-Hurwitz formula gives total ramification $\sum (e_p-1)=6$. The number of distinct ramification points requires a further distinction. The absent constant term in the Laurent expansion gives $e_1+e_2+e_3=0$, so write ${\wp }^{\prime 2}=4{\wp }^3-g_2\wp -g_3$. Differentiation yields ${\wp }^{\prime \prime }=6{\wp }^2-g_2/2$. If $g_2\ne 0$, each of the two values $\pm \sqrt{g_2/12}$ has two distinct preimages under $\wp$ and none is a branch value, since a common zero of the cubic and its derivative would be a repeated cubic root. Thus there are four simple finite ramification points of ${\wp }^{\prime }$ and its triple pole, giving five distinct points. If $g_2=0$, the nonsingularity of the cubic forces $g_3\ne 0$. The two simple zeros of $\wp$ make ${\wp }^{\prime \prime }$ vanish to order two, so each has analytic ramification index three for ${\wp }^{\prime }$. Together with its triple pole there are three distinct points. This second case occurs for a hexagonal lattice: its sixth-root rotational symmetry forces the $z^2$ Laurent coefficient, and hence $g_2$, to vanish. The weighted count is six in both cases.

For a degree-$d$ nonconstant holomorphic map from a compact connected Riemann surface of genus $g$ to the projective line, the Riemann-Hurwitz formula iswhere $e_p$ is the local degree. A meromorphic function defines a map to the sphere by giving its ordinary finite values away from poles and value infinity at each pole. The reciprocal function is holomorphic in a neighborhood of a pole, proving holomorphicity there. For use in Riemann-Hurwitz the function must be nonconstant; a nonzero constant still defines a holomorphic map, but of degree zero.

Let $X=\mathbb{C}/\Lambda$. The double pole of $\wp$ gives a degree-two map to the sphere. The torus has genus one, so total ramification is four. One simple ramification point is zero, above infinity. At the three nonzero half-period classes $h_i$, evenness and periodicity give ${\wp }^{\prime }(h_i)=0$. Each is a ramification point of local degree two, and these account for all remaining ramification. Put $e_i=\wp (h_i)$. They are distinct, since a degree-two fiber cannot contain two points each of local degree two.

At each $h_i$, $\wp -e_i$ has a double zero and ${\wp }^{\prime }$ has a simple zero. Consequently the elliptic quotienthas removable singularities there and no zeros or poles elsewhere except possibly zero. Near zero, $\wp =z^{-2}+O(z^2)$ and ${\wp }^{\prime }=-2z^{-3}+O(z)$, so $R\to 4$. Thus $R$ is holomorphic on the compact torus and constant, provingThe printed additional requirement that all three $e_i$ be nonzero is false. For the square lattice $\Lambda =\mathbb{Z}+i\mathbb{Z}$, invariance of the defining series under multiplication by $i$ gives $\wp (iz)=-\wp (z)$. At $h=(1+i)/2$, one has $ih=h-1$, so periodicity gives $\wp (h)=\wp (ih)=-\wp (h)$ and hence $\wp (h)=0$. This is one of the three branch values. Distinctness, which is all the cubic construction needs, remains valid.

DefineThe image lies on $Y^{2}Z=4(X-e_{1}Z)(X-e_{2}Z)(X-e_{3}Z)$. Distinct roots make its affine part smooth, and the point at infinity is smooth because the derivative of its equation with respect to $Z$ is nonzero there. Near zero the projective coordinates $X/Y=\wp /{\wp }^{\prime }\sim -z/2$ and $Z/Y=1/{\wp }^{\prime }\sim -z^3/2$ extend holomorphically. Every finite $X/Z$ value has two preimages under $\wp$ except at a branch value. Evenness pairs those preimages as $z,-z$, and oddness of ${\wp }^{\prime }$ distinguishes them unless they are the same half-period class. Therefore the two coordinates together give injectivity and reach both signs of $Y$, proving surjectivity onto the cubic. Thus $\Phi$ is a bijective holomorphic map to a nonsingular cubic, including when one branch value is zero.

The given Weierstrass elliptic function is even, by replacing the lattice index $\omega$ with $-\omega$. Differentiation givesFor any $\lambda \in \Lambda$, reindexing the sum shows ${\wp }^{\prime }(z+\lambda )={\wp }^{\prime }(z)$. Thus $\wp (z+\lambda )-\wp (z)=c_{\lambda }$ is constant off the lattice. Replacing $z$ by $-z-\lambda$ and using evenness changes this difference to its negative, so $c_{\lambda }=-c_{\lambda }$ and $c_{\lambda }=0$. Hence $\wp$ is doubly periodic.

It therefore descends to a meromorphic function on $\mathbb{C}/\Lambda$, with a double pole at zero modulo the lattice and no others. A meromorphic function is a holomorphic map to the Riemann sphere, sending poles to infinity. Locally at zero, $1/\wp (z)=z^2+O(z^4)$ is holomorphic and vanishes there, so the sphere chart at infinity supplies the required extension. The map is nonconstant because it has a pole.

For a nonconstant holomorphic map of degree $d$ between compact connected Riemann surfaces, the Riemann-Hurwitz formula isHere $e_p$ is the local degree, characterized by local coordinates in which the map is $z\mapsto z^{e_p}$. The sum is finite and its terms are nonnegative; a ramification point has $e_p>1$.

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, let $n$ be its degree and $e_p$ its local degree at $p$. The Riemann-Hurwitz formula isEquivalently, $2g_X-2=n(2g_Y-2)+{\sum }_p(e_p-1)$. Only finitely many terms are nonzero; their sum is the total ramification. In local coordinates the map has the form $w=z^{e_p}$ after suitable coordinate changes.

Let $x\in X$ and choose holomorphic coordinate charts centred at $x$ and $f(x)$. A nonconstant holomorphic map has a local expansion $w\circ f(z)=c{z}^m+O(z^{m+1})$, where $c\ne 0$ and $m\ge 1$. Taking a local holomorphic $m$th root of the nonvanishing factor and changing the source coordinate gives the exact local form $w=z^m$. The integeris the ramification index of a holomorphic map. A ramification point of a holomorphic map is a source point with ${\nu }_f(x)>1$, equivalently where its derivative vanishes; a branch value of a holomorphic map, also called a branch point here, is the image of such a point in $Y$. These are different sets, one in the source and one in the target.

Every fiber of a function is finite: locally it is the zero set of a nonzero holomorphic function, so its points are isolated, and compactness prevents infinitely many such zeros without an accumulation point. The same argument applied to the local derivatives makes the set of ramification points of a holomorphic map finite. The open mapping theorem and compactness show that $f(X)$ is both open and closed in the connected target $Y$, hence $f$ is surjective.

For any $y\in Y$, the degree of a holomorphic map isTo see independence of $y$, choose disjoint small coordinate charts about its finitely many preimages. The local form $z\mapsto z^{{\nu }_f(x)}$ shows that nearby values have ${\nu }_f(x)$ preimages counted with multiplicity in the corresponding chart. Compactness excludes additional preimages outside these charts for sufficiently nearby values: a sequence of such preimages would have a limit in $f^{-1}(y)$. The displayed sum is therefore locally constant, hence constant on connected $Y$. Away from branch values of a holomorphic map it is the ordinary number of preimages, and $f$ is a $d$-sheeted covering map there.

The Riemann-Hurwitz formula isor, in terms of the two genera,For its proof, choose a triangulation of a surface on $Y$ containing every branch value of a holomorphic map among its vertices. Every open edge and open face has $d$ lifts, since its interior avoids branch values of a holomorphic map. The lifted cells give a finite CW complex structure on $X$. If $V_Y,E_Y,F_Y$ are the target counts and $R={\sum }_x({\nu }_f(x)-1)$, the source counts areIndeed, at a target vertex the deficit from $d$ distinct preimages is precisely ${\sum }_{x\mapsto y}({\nu }_f(x)-1)$, and all nonzero contributions occur over vertices. Taking the alternating sum gives $\chi (X)=d(V_Y-E_Y+F_Y)-R$, as required. This is the Triangulation proof of the Riemann-Hurwitz formula.

The printed proximity integral is missing a factor $1/(2\pi )$. Denote that literal integral by $m_{\mathrm{print}}$ and write $\overline{m}=m_{\mathrm{print}}/(2\pi )$. The correct Annular spherical Jensen formula, with the printed definitions of $N$ and $T$, isThe following derivation proves this identity and also isolates the normalization issue.

For the chordal metric put $u_a(z)=-\log k(f(z),a)$ and let $M_a(s)=(2\pi {)}^{-1}{\int }_0^{2\pi }{u}_a(s{e}^{i\theta })d\theta$. In a finite target coordinate,For $a=\infty$ use $u_{\infty }=\frac{1}{2}\log (1+|f{|}^2)$. Differentiating away from the preimages of $a$ gives half the squared spherical derivative of a meromorphic function. At an $a$-point of local degree $d$, $u_a=-d\log |z-z_0|+O(1)$; the second distributional derivatives give the Laplacian of $\log |z-z_0|$ is $2\pi {\delta }_{z_0}$. Thus, including poles by changing spherical coordinates,Because $a$ is off the image of $|z|=1$, $u_a$ is smooth near that circle and $I(a)=M_a^{\prime }(1)$ is finite and independent of $r$.

Apply Green's second identity to the harmonic weight $w(z)=\log (|z|/r)$ on $r<|z|<1$. On the outer boundary $w=\log (1/r)$ and ${\partial }_{n}w=1$; on the inner boundary $w=0$ and ${\partial }_{n}w=-1/r$. HenceThe distributional formula makes the left side $2\pi T(r)-2\pi N(r;a)$, and $M_a(r)-M_a(1)=\overline{m}(r;a)$, proving the corrected Annular spherical Jensen formula. Initially take an inner circle avoiding $a$-points. Logarithmic singularities have continuous angular means, and a point on the inner circle has zero logarithmic counting weight, so continuity extends the identity to every $0<r\le 1$.

There is no way to absorb the missing factor into a different constant $I(a)$. For the holomorphic function $f(z)=z$ and $a=\infty$, direct integration givesThe value of $(T-m_{\mathrm{print}})/\log (1/r)$ tends to $1/2$ as $r\downarrow 0$ and to $\pi$ as $r\uparrow 1$. Therefore the literal printed identity has no constant $I(\infty )$, whereas the corrected identity has $I(\infty )=1/2$.

For the singularity criterion, define the nondecreasing normalized spherical areaTonelli's theorem applied to the logarithmic weight gives $T(e^{-l})={\int }_0^{l}s(t)dt$. Averages of a nondecreasing function converge to its supremum: for fixed $t_0<l$, $(1-t_0/l)s(t_0)\le T(e^{-l})/l\le s(l)$. Thus the limit, not just the lower limit, exists in $[0,\infty ]$ and equalsIf $f$ has a removable singularity or pole at zero, it extends as a holomorphic map to the Riemann sphere; the pulled-back spherical density is smooth and bounded on the closed disk, hence its area is finite. Conversely an essential singularity gives, by the meromorphic version of the Great Picard theorem, infinitely many preimages of every spherical target with at most two exceptions in each punctured neighborhood. The area formula givesThe formula follows by applying change of variables on inverse-function neighborhoods away from the discrete critical points, then summing with multiplicity; the countable critical values have area zero. This proves the finite spherical area criterion for an isolated singularity and the requested equivalence between a finite lower limit and a removable singularity or pole.