Geometric Langlands for GL_1

At the forefront of contemporary mathematics, the Langlands Program—initiated by Robert Langlands in the late 1960s—seeks to relate two seemingly distant worlds. On one side are Galois representations, which encode the symmetries of algebraic equations through continuous homomorphisms from absolute Galois groups into matrix groups. On the other side are automorphic forms, highly symmetric analytic functions on the adele group of a reductive algebraic group, generalizing classical modular forms. The central conjecture posits a correspondence between these objects: to each $\displaystyle n$ -dimensional Galois representation, there should exist an automorphic representation carrying equivalent arithmetic information. This vision organizes number theory, representation theory, and algebraic geometry into a single framework capable of translating problems—and their solutions—across disciplines.

The original Langlands correspondence was formulated over global fields of characteristic zero, such as the number field $\displaystyle \mathbb{Q}$ , where one studies the relationship between $\displaystyle n$ -dimensional $\displaystyle \ell$ -adic Galois representations of $\displaystyle \mathrm{Gal}(\overline{F}/F)$ and automorphic representations of reductive groups over $\displaystyle F$ . Owing to the formidable arithmetic difficulties in this setting, a geometric variant arises by replacing the global field $\displaystyle F$ with the function field of a smooth, projective curve $\displaystyle X$ over $\displaystyle \mathbb{C}$ . In this context, the absolute Galois group is replaced by the topological fundamental group $\displaystyle \pi_1(X)$ , and the relevant structures can be recast in the language of algebraic geometry, perverse sheaves, and $\displaystyle \mathcal{D}$ -modules, with the Riemann surface underlying $\displaystyle X$ providing the natural geometric stage for the correspondence.

The rank-1 case, corresponding to the group $\displaystyle GL_1$ , is completely understood and serves as a prototype for the far more intricate non-abelian theories. Here, the correspondence manifests as a duality—reminiscent of a geometric Fourier transform—between the moduli of rank-1 local systems on a curve (encoding monodromy representations of loops) and the Picard variety (parametrizing holomorphic line bundles). This equivalence distills the essence of geometric Langlands into its most transparent form.

The geometric Langlands correspondence unfolds on the stage of Riemann surfaces—one-dimensional complex manifolds equipped with an atlas whose transition maps are holomorphic. Topologically, such surfaces are classified by their genus $\displaystyle g$ (number of handles) and number of punctures, data which fully determine their orientable structure. A complex structure endows each point with local coordinates in $\displaystyle \mathbb{C}$ , enabling the machinery of complex analysis to operate intrinsically, independent of any embedding in $\displaystyle \mathbb{R}^3$ . Historically, Riemann surfaces arose as the natural domains for multi-valued analytic functions such as $\displaystyle \sqrt{z}$ or $\displaystyle \log z$ . By constructing branched coverings of $\displaystyle \mathbb{C}$ , one resolves branch points into globally well-defined holomorphic structures. Thus, the analytic singularities of a function dictate the topology and geometry of the surface on which it becomes single-valued.

The “Spectral” Side: Loops, Monodromy, and Local Systems

In the rank-1 geometric Langlands correspondence, the spectral side is built from the topology of the underlying curve (or Riemann surface) $\displaystyle X$ . Here, the topological invariants are encoded in the fundamental group, which classifies loops up to continuous deformation. This group encodes how loops on $\displaystyle X$ can wind around its “holes” and other features.

The Fundamental Group:

Let $\displaystyle X$ be a surface and choose a base point $\displaystyle p \in X$ .

Loops: A loop based at $\displaystyle p$ is a continuous path that starts and ends at $\displaystyle p$ .
Homotopy: Two loops are homotopically equivalent if one can be continuously deformed into the other without tearing the surface or moving the base point. Think of a loop as a rubber band: if you can slide and stretch it into another without breaking it, the two are equivalent.
The Group $\displaystyle \pi_1(X,p)$ :
- Operation: Concatenate two loops to form a new loop.
- Identity: The constant loop at $\displaystyle p$ (doing nothing).
- Inverse: The same loop traversed in the opposite direction.

If $\displaystyle X$ is path-connected, the group is well-defined up to isomorphism regardless of the choice of $\displaystyle p$ , so we often write simply $\displaystyle \pi_1(X)$ .

Examples:

Sphere $\displaystyle S^2$ : On a sphere, any rubber band can be slid and shrunk until it’s just a single point. So, every loop is homotopically equivalent to the constant loop. Every loop can be contracted to a point, so the fundamental group $\displaystyle \pi_1(S^2) \cong \{e\}$ the trivial group.
Circle $\displaystyle S^1$ : You can walk around a circle once, or twice, or three times. Each of these loops is distinct. You can also walk around it in the opposite direction. The number of times you wind around is an integer, positive or negative. So the loops are classified by their winding number (an integer), so $\displaystyle \pi_1(S^1) \cong \mathbb{Z}$
Torus $\displaystyle T^2$ : There are two basic, non-equivalent loops: one that goes through the hole of the donut (the “meridian” loop) and one that goes around the tube of the donut (the “longitude” loop). Any other loop can be formed by combining these two in some way, so $\displaystyle \pi_1(T^2) \cong \mathbb{Z}^2$ .
Punctured Plane $\displaystyle \mathbb{C} \setminus {0}$ : : The complex plane with the origin removed is topologically a cylinder, which is equivalent to a circle. The only non-trivial loops are those that wind around the origin. So $\displaystyle \pi_1(\mathbb{C} \setminus \{0\}) \cong \mathbb{Z}$ .

Monodromy

We often assume functions are “well-behaved” and single-valued. But what if a function’s value depends on the path taken to get there? This is the core idea of monodromy that loops can change the value of a function, even when the loop returns to its starting point.

Example 1: The Complex Logarithm
On $\displaystyle X = \mathbb{C} \setminus {0}$ , $\displaystyle \log z = \ln |z| + i\, \mathrm{arg}(z)$ is multi-valued because $\displaystyle \mathrm{arg}(z)$ is defined only up to $\displaystyle 2\pi k$ , $\displaystyle k \in \mathbb{Z}$ . The fundamental group of $\displaystyle X = \mathbb{C} \setminus {0}$ is $\displaystyle \mathbb{Z}$ , generated by one counterclockwise loop around the origin. If we analytically continue $\displaystyle \log z$ around this loop, the value changes by

$\displaystyle \log z \mapsto \log z + 2\pi i$ .

This is the monodromy transformation: going around the origin once “adds” $\displaystyle 2\pi i$ to the value.

Example 2: Square Root
Similarly, for $\displaystyle f(z) = \sqrt{z}$ , start at $\displaystyle z_0 = 1$ with $\displaystyle f(1) = 1$ . A counterclockwise loop $\displaystyle \gamma$ around the origin increases the argument $\displaystyle \arg(z)$ by $\displaystyle 2\pi$ . Since $\displaystyle \sqrt{z} = |z|\, e^{i\,\arg(z)/2}$ , this change sends $\displaystyle e^{i(0/2)} = 1 \quad\longmapsto\quad e^{i(2\pi/2)} = e^{i\pi} = -1$ . Going around a second time adds another $\displaystyle 2\pi$ to the argument, for a total change of $\displaystyle 4\pi$ : $\displaystyle e^{i(4\pi/2)} = e^{i 2\pi} = 1$ . Thus, the monodromy is multiplicative and of order 2: each loop around the origin multiplies the value by $\displaystyle -1$ , and two loops restore the original value.

Local Systems: From Monodromy to Representations

A local system is a formal way to describe how a function or object can “twist” as it’s transported along paths. It’s a geometric object that is “locally trivial” (like a simple vector space) but can be “globally twisted” by topology.

A rank-1 local system assigns a one-dimensional complex vector space (a copy of C) to each point on the curve X. We then need a rule for how to move a vector from the space at one point to the space at a nearby point. This rule is called a flat connection.

Connection: A connection is a rule for “parallel transport.” It tells us how to move a vector along a path while keeping it “pointing in the same direction” in a certain sense.

Flatness: A connection is flat if parallel transport around any loop that can be shrunk to a point leaves the vector unchanged. This means the result of parallel transport depends only on the homotopy class of the loop, not the specific path taken. So the flatness condition ensures that if a vector is transported along a path and then back to its starting point—without encircling any topological “holes” of $\displaystyle X$ —it returns unchanged.

If the path $\displaystyle \gamma$ is a loop (starting and ending at the same point), the parallel transport along $\displaystyle \gamma$ induces a linear transformation of the fiber at that point. This transformation is, by definition, the monodromy of $\displaystyle \gamma$ .

In the rank-1 case, each fiber is a one-dimensional complex vector space, naturally identifiable with $\displaystyle \mathbb{C}$ . Any linear transformation of such a space is simply multiplication by a complex scalar. Because parallel transport is always reversible, these scalars must be nonzero—elements of $\displaystyle \mathbb{C}^\times$ .

Thus, for each loop $\displaystyle \gamma \in \pi_1(X)$ , parallel transport yields a monodromy factor $\displaystyle \rho(\gamma) \in \mathbb{C}^\times$ . The fundamental group $\displaystyle \pi_1(X)$ encodes the loop structure of $\displaystyle X$ , and the concatenation of loops corresponds to the composition of monodromy transformations. Consequently,

$\displaystyle \rho : \pi_1(X) \to \mathbb{C}^\times$

must be a group homomorphism.

This homomorphism $\displaystyle \rho$ is the algebraic essence of a rank-1 local system. The collection of all such homomorphisms, $\displaystyle \mathrm{Hom}(\pi_1(X),\mathbb{C}^\times)$ forms the moduli space of rank-1 local systems, denoted $\displaystyle \mathrm{Loc}_1(X)$ .

Punctured Complex Plane ( $\displaystyle X = \mathbb{C}^\times$ )

Fundamental Group: $\displaystyle \pi_1(\mathbb{C}^\times)$ is the infinite cyclic group, isomorphic to $\displaystyle \mathbb{Z}$ . It is generated by a single loop $\displaystyle \gamma_0$ that winds counterclockwise once around the origin. Any other loop is equivalent to traversing $\displaystyle \gamma_0$ an integer number of times.
Rank-1 Local Systems:
A homomorphism $\displaystyle \rho: \mathbb{Z} \to \mathbb{C}^\times$ is determined by the image of $\displaystyle 1 \in \mathbb{Z}$ .
If $\displaystyle \rho(1) = \lambda \in \mathbb{C}^\times$ , then $\displaystyle \rho(n) = \lambda^n$ .
Moduli Space:
Each $\displaystyle \lambda \in \mathbb{C}^\times$ corresponds to a unique local system. $\displaystyle \mathrm{Loc}_1(\mathbb{C}^\times) \cong \mathbb{C}^\times$

Torus ( $\displaystyle X = T^2$ )

Fundamental Group:
$\displaystyle \pi_1(T^2) \cong \mathbb{Z} \times \mathbb{Z}$ , generated by $\displaystyle \gamma_a$ (around the “hole”) and $\displaystyle \gamma_b$ (around the “tube”).
Rank-1 Local Systems:
A homomorphism $\displaystyle \rho: \mathbb{Z}^2 \to \mathbb{C}^\times$ is determined by $\displaystyle \rho(\gamma_a) = \lambda_a$ , $\displaystyle \rho(\gamma_b) = \lambda_b$ .
Moduli Space:
Specified by pairs $\displaystyle (\lambda_a, \lambda_b) \in \mathbb{C}^\times \times \mathbb{C}^\times$ . $\displaystyle \mathrm{Loc}_1(T^2) \cong \mathbb{C}^\times \times \mathbb{C}^\times$

Surface with $\displaystyle g$ Holes ( $\displaystyle \Sigma_g$ )

Fundamental Group:
$\displaystyle \pi_1(\Sigma_g)$ is non-abelian with $\displaystyle 2g$ generators.
Rank-1 Local Systems:
Since $\displaystyle \mathbb{C}^\times$ is abelian, any homomorphism $\displaystyle \rho$ factors through the abelianization of $\displaystyle \pi_1(\Sigma_g)$ , which is $\displaystyle \mathbb{Z}^{2g}$ .
Moduli Space:
Determined by the images of the $\displaystyle 2g$ generators of the abelianized group: $\displaystyle \mathrm{Loc}_1(\Sigma_g) \cong (\mathbb{C}^\times)^{2g}$

Flat Connection

The interpretation of a local system as a vector bundle with a flat connection becomes especially concrete in the rank-1 case when expressed in terms of differential forms.

A flat connection on a rank-1 bundle can be described by a globally defined closed $\displaystyle 1$ -form $\displaystyle \omega$ . For a loop $\displaystyle \gamma$ , the monodromy is given by

$\displaystyle \rho(\gamma) = \exp\!\left( \int_\gamma \omega \right)$

We can now revisit our earlier examples from this perspective.

The fundamental group is $\displaystyle \pi_1(\mathbb{C}^\times) \cong \mathbb{Z}$ generated by a single counterclockwise loop $\displaystyle \gamma_0$ around the origin.

Consider the differential $\displaystyle 1$ -form $\displaystyle \omega = \frac{dz}{z}$ This form is closed on $\displaystyle \mathbb{C}^\times$ since $\displaystyle d\omega = d(dz/z) = 0$ , so it is a valid connection form for a flat line bundle, because the integral along contractible loops evaluate to zero by Stokes theorem.

We parametrize $\displaystyle \gamma_0$ as a circle of radius $\displaystyle r$ centered at the origin: $\displaystyle z(t) = r e^{it}, \quad 0 \le t \le 2\pi$ . The derivative is $\displaystyle dz = i r e^{it} dt$ , so

$\displaystyle \int_{\gamma_0} \omega = \int_{\gamma_0} \frac{dz}{z} = \int_0^{2\pi} \frac{i r e^{it}}{r e^{it}} \, dt = \int_0^{2\pi} i \, dt = 2\pi i$

This $\displaystyle 2\pi i$ measures the total “twist” of the connection around the puncture. The monodromy factor is the exponential:

$\displaystyle \rho(\gamma_0) = \exp\left( \int_{\gamma_0} \omega \right) = \exp(2\pi i) = 1$

Thus, parallel transport around $\displaystyle \gamma_0$ leaves every fiber unchanged. This is the trivial rank-1 local system.

To get a non-trivial local system, choose $\displaystyle \omega = c\, \frac{dz}{z}, \quad c \in \mathbb{C}$ . This is also closed, hence defines a flat connection. Its integral around $\displaystyle \gamma_0$ is

$\displaystyle \int_{\gamma_0} \omega = c \int_{\gamma_0} \frac{dz}{z} = c(2\pi i) = 2\pi i c$

The corresponding monodromy is $\displaystyle \rho(\gamma_0) = \exp(2\pi i c)$

By varying $\displaystyle c$ , we obtain a wide range of non-trivial holonomies in $\displaystyle \mathbb{C}^\times$ .

Consider $\displaystyle f(z) = \sqrt{z}$ . As we saw earlier, going once counterclockwise around the origin multiplies $\displaystyle f(z)$ by $\displaystyle -1$ . Our goal is to find a connection form whose monodromy reproduces this multiplicative factor. We want $\displaystyle \exp(2\pi i\, c) = -1$ . A simple solution is $\displaystyle c = \frac{1}{2}$ . Thus, the connection form for the corresponding local system is $\displaystyle \omega = \frac{1}{2} \frac{dz}{z}$ . The integral around the generator $\displaystyle \gamma_0$ of $\displaystyle \pi_1(\mathbb{C}^\times)$ is

$\displaystyle \int_{\gamma_0} \omega = \frac{1}{2} \int_{\gamma_0} \frac{dz}{z} = \frac{1}{2} (2\pi i) = \pi i$

Exponentiating gives $\displaystyle \rho(\gamma_0) = \exp(\pi i) = -1$ This local system encodes exactly the “twist” of the square root: a multiplicative monodromy of order $\displaystyle 2$ . While the function’s analytic continuation changes sign, the local system captures this change as the holonomy of a flat connection.

If we want a unit-circle monodromy $\displaystyle \lambda = e^{i\theta}$ , choose $\displaystyle c = \frac{\theta}{2\pi}$ and set $\displaystyle \omega = \frac{\theta}{2\pi} \frac{dz}{z}$ . Then $\displaystyle \int_{\gamma_0} \omega = i\theta,\quad \rho(\gamma_0) = e^{i\theta}$ . Varying $\displaystyle \theta$ gives any desired rotation.

To summarize: Any $\displaystyle \lambda \in \mathbb{C}^\times$ defines a rank-1 local system on $\displaystyle \mathbb{C}^\times$ via $\displaystyle \rho(n) = \lambda^n,\quad n \in \mathbb{Z}$ . This can be realized by the flat connection $\displaystyle \omega = c\,\frac{dz}{z}$ with $\displaystyle c \in \mathbb{C}$ satisfying $\displaystyle \lambda = e^{2\pi i c}$ . A convenient choice is $\displaystyle c = \frac{1}{2\pi i} \log\lambda$ where the branch of $\displaystyle \log$ is fixed. Thus, every rank-1 local system on the punctured plane arises from such a simple differential form.

The “Automorphic” Side: Line Bundles and the Picard Group

The second side of the Langlands duality, often called the “automorphic” side, is concerned not with the loops within the surface X, but with the space of all possible ways to attach a complex line to each point of X. This leads to the concepts of line bundles and their classifying space, the Picard group.

Attaching a Line to Every Point: Line Bundles

A line bundle over a Riemann surface $\displaystyle X$ is a space $\displaystyle L$ equipped with a map $\displaystyle \pi : L \to X$ such that for every point $\displaystyle p \in X$ , the fiber $\displaystyle \pi^{-1}(p)$ is a one-dimensional complex vector space (a complex line). The key condition is local triviality: for any sufficiently small open set $\displaystyle U \subset X$ , the portion of the bundle over it, $\displaystyle \pi^{-1}(U)$ , is isomorphic to the simple product $\displaystyle U \times \mathbb{C}$ . However, globally the bundle can be twisted. A classic example of a non-trivial real line bundle is the Möbius strip, built over a circle $\displaystyle S^1$ . Locally, any small piece of the Möbius strip looks like a rectangle (a piece of the circle times an interval). But globally, it has a twist: traversing the entire circle returns you to the same fiber with its orientation flipped. A line bundle that is globally a product, $\displaystyle L \cong X \times \mathbb{C}$ is called the trivial line bundle.

The Universe of Line Bundles: The Picard Group

The collection of all line bundles on a surface $\displaystyle X$ has a rich algebraic structure. We regard two line bundles as the same if they are isomorphic. The set of isomorphism classes forms an abelian group called the Picard group, denoted $\displaystyle \mathrm{Pic}(X)$

Group Operation — Tensor Product: Given two line bundles $\displaystyle L_1$ and $\displaystyle L_2$ , their tensor product $\displaystyle L_1 \otimes L_2$ is a new line bundle whose fiber over a point $\displaystyle p$ is

$\displaystyle (L_1 \otimes L_2)_p = L_{1,p} \otimes L_{2,p}$

Identity Element: he identity element is the trivial line bundle

$\displaystyle \mathcal{O}_X = X \times \mathbb{C}$

Inverse Element: The inverse of a line bundle $\displaystyle L$ is its dual bundle

$\displaystyle L^\vee = \mathrm{Hom}(L, \mathcal{O}_X)$

whose fiber over $\displaystyle p$ consists of linear maps from $\displaystyle L_p$ to $\displaystyle \mathbb{C}$ .

While the previous explanation of line bundles is intuitive, a more rigorous and insightful way to understand them is through transition maps. This perspective is fundamental for actual calculations.

A line bundle $\displaystyle L$ over a surface $\displaystyle X$ is a space where a copy of the complex line $\displaystyle \mathbb{C}$ is “attached” to every point of $\displaystyle X$ . The key idea is that this attachment is done locally. We cover $\displaystyle X$ with open sets $\displaystyle {U_i}$ . Over each open set $\displaystyle U_i$ , the line bundle is trivial, meaning it looks like the product $\displaystyle L|_{U_i} \cong U_i \times \mathbb{C}$ . A local trivialization is a map giving this isomorphism, letting us describe the fibers over $\displaystyle U_i$ in coordinates. A section of the line bundle is a continuous choice of a vector in each fiber. Locally, a section over $\displaystyle U_i$ can be written as a function $\displaystyle s_i(p) \in \mathbb{C}, \quad p \in U_i$ . The twist of the bundle is revealed when we compare these local descriptions on overlapping patches: the transition functions between trivializations encode the bundle’s global geometry.

The Role of Transition Maps

Let $\displaystyle U_i$ and $\displaystyle U_j$ be overlapping open sets in $\displaystyle X$ . Each provides a local trivialization of the line bundle $\displaystyle L$ . On their intersection $\displaystyle U_i \cap U_j$ , the same vector in a fiber can be expressed in either coordinate system. The transition map

$\displaystyle g_{ij} : U_i \cap U_j \to \mathbb{C}^\times$

converts between these two local descriptions: if $\displaystyle s_i$ is a local section over $\displaystyle U_i$ , then over $\displaystyle U_j$ the corresponding section satisfies

$\displaystyle s_j(p) = g_{ij}(p) \cdot s_i(p), \quad p \in U_i \cap U_j$

Here $\displaystyle g_{ij}(p)$ is a nonzero complex number giving the “twist” between the trivializations. On a triple overlap $\displaystyle U_i \cap U_j \cap U_k$ , compatibility of all three local descriptions imposes the cocycle condition:

$\displaystyle g_{ik}(p) = g_{ij}(p) \cdot g_{jk}(p), \quad p \in U_i \cap U_j \cap U_k$

The entire line bundle is determined (up to isomorphism) by the collection of transition functions $\displaystyle {g_{ij}}$ satisfying this cocyle condition.

The transition map formalism makes the group structure of the Picard group completely explicit.

Tensor Product $\displaystyle (L_1 \otimes L_2)$ :
If $\displaystyle L_1$ and $\displaystyle L_2$ are defined by transition maps $\displaystyle {g^{(1)}{ij}}$ and $\displaystyle {g^{(2)}{ij}}$ , then their tensor product has transition maps

$\displaystyle g_{ij}(p) = g^{(1)}_{ij}(p) \cdot g^{(2)}_{ij}(p)$

Fiberwise multiplication of transition maps exactly matches the tensor product of the line bundles.

Identity Element $\displaystyle (\mathcal{O}_X)$ :
The trivial bundle is defined by $\displaystyle g_{ij}(p) = 1$

Multiplying any transition functions by $\displaystyle 1$ leaves them unchanged, so this is the identity in $\displaystyle \mathrm{Pic}(X)$ .

Inverse Element $\displaystyle (L^\vee)$ :
If $\displaystyle L$ has transition maps $\displaystyle {g_{ij}}$ , then its dual bundle $\displaystyle L^\vee$ has transition maps $\displaystyle g^\vee_{ij}(p) = \frac{1}{g_{ij}(p)}.$

The tensor product $\displaystyle L \otimes L^\vee$ then has transition maps $\displaystyle g_{ij}(p) \cdot \frac{1}{g_{ij}(p)} = 1$ so it is the trivial bundle.

Line Bundles on the Riemann Sphere ( $\displaystyle \mathbb{P}^1$ )

The Riemann sphere $\displaystyle \mathbb{P}^1$ is topologically a sphere. It has a single non-trivial transition map gluing together two coordinate charts:

$\displaystyle U_0$ with coordinate $\displaystyle z$
$\displaystyle U_1$ with coordinate $\displaystyle w$

On their overlap $\displaystyle \mathbb{C}^\times$ , the coordinates satisfy $\displaystyle w = 1/z$ .

A line bundle on $\displaystyle \mathbb{P}^1$ is completely determined by a transition map of the form:

$\displaystyle g_{01}(z) = z^k$

where $\displaystyle k \in \mathbb{Z}$ is the degree of the line bundle.

Degree $\displaystyle k = 0$ — The Trivial Line Bundle $\displaystyle \mathcal{O}_{\mathbb{P}^1}$ . The transition map is: $\displaystyle g_{01}(z) = z^0 = 1$ . This is the simplest bundle, where the local pieces are glued with no twist. A global section is a function that is holomorphic everywhere on the sphere. The only such functions are the constants.

Degree $\displaystyle k = 1$ — The Hyperplane Bundle $\displaystyle \mathcal{O}(1)$ . The transition map is: $\displaystyle g_{01}(z) = z$ . A section on $\displaystyle U_0$ is given by a function $\displaystyle f_0(z)$ , and a section on $\displaystyle U_1$ by a function $\displaystyle f_1(w)$ . On the overlap, the gluing condition is $\displaystyle f_0(z) = z \cdot f_1(1/z)$ . We have two global sections $\displaystyle 1$ and $\displaystyle z$ — spanning a 2-dimensional space of sections (linear polynomials).

Degree $\displaystyle k = -1$ — The Tautological Bundle $\displaystyle \mathcal{O}(-1)$ . The transition map: $\displaystyle g_{01}(z) = z^{-1}$ . Gluing condition becomes $\displaystyle f_0(z) = z^{-1} f_1(1/z)$ . The only possible sections are the zero section. This bundle is often described as the line at each point that passes through the origin, that describes the point.(Recall that the point of $\displaystyle \mathbb{P}^1$ are lines through origin.)

Degree $\displaystyle k = -2$ — The Canonical Bundle $\displaystyle \mathcal{O}(-2)$ . Transition map: $\displaystyle g_{01}(z) = z^{-2}$ . Gluing condition: $\displaystyle f_0(z) = z^{-2} f_1(1/z)$ . No nonzero global sections on this bundle. The space is related to differential forms: a global holomorphic $\displaystyle 1$ -form corresponds to a section of this bundle, and none exist on $\displaystyle \mathbb{P}^1$ .

Line Bundles on the Torus

The complex torus is obtained from the complex plane $\displaystyle \mathbb{C}$ by identifying points that differ by an integer linear combination of $\displaystyle 1$ and a fixed complex number $\displaystyle \tau$ with $\displaystyle \mathrm{Im}(\tau) > 0$ . Formally, $\displaystyle T^2 \;=\; \mathbb{C} / (\mathbb{Z} + \mathbb{Z}\tau)$ We can picture this as a fundamental parallelogram in the complex plane, with vertices $\displaystyle 0,, 1,, \tau,, 1+\tau$ , whose opposite edges are identified.

A convenient open cover of the torus consists of four overlapping open sets:

$\displaystyle U_{00}$ centered at $\displaystyle 0$
$\displaystyle U_{10}$ centered at $\displaystyle 1$
$\displaystyle U_{01}$ centered at $\displaystyle \tau$
$\displaystyle U_{11}$ centered at $\displaystyle 1+\tau$

Identifications occur along the edges:

The top edge of $\displaystyle U_{00}$ is identified with the bottom edge of $\displaystyle U_{01}$ via a shift by $\displaystyle \tau$ .
The right edge of $\displaystyle U_{00}$ is identified with the left edge of $\displaystyle U_{10}$ via a shift by $\displaystyle 1$ .

The torus is formed by identifying $\displaystyle U_{00}$ with $\displaystyle U_{10}$ (via the transition $\displaystyle z \mapsto z - 1$ ) and with $\displaystyle U_{01}$ (via the transition $\displaystyle z \mapsto z - \tau$ ). The transition functions tell us how to glue the fibers over the overlaps. A line bundle is a pair of local sections $\displaystyle (s_{00}, s_{10})$ on the overlap, where $\displaystyle s_{00}$ is defined with respect to the trivialization on $\displaystyle U_{00}$ and $\displaystyle s_{10}$ with respect to $\displaystyle U_{10}$ . The transition function $\displaystyle g_{00,10}(z)$ on the overlap $\displaystyle U_{00} \cap U_{10}$ tells us that $\displaystyle s_{10}(z - 1) = g_{00,10}(z) \, s_{00}(z)$ .

Trivial Bundle (Degree $\displaystyle 0$ ):

We need to specify the transition functions for all the overlaps. The easiest way to get the trivial bundle is to make all transition functions equal to 1.

Overlap of $\displaystyle U_{00}$ and $\displaystyle U_{10}$ : $\displaystyle g_{00,10}(z) = 1$
Overlap of $\displaystyle U_{00}$ and $\displaystyle U_{01}$ : $\displaystyle g_{00,01}(z) = 1$

The cocycle condition holds trivially. For example, on the triple overlap $\displaystyle U_{00} \cap U_{10} \cap U_{01}$ , we have $\displaystyle g_{00,01}(z) = g_{00,10}(z) g_{10,01}(z)$ which becomes $\displaystyle 1 = 1 \cdot 1$ .

A global section is a collection of local sections that agree on the overlaps. Since all transition functions are 1, all local sections must be the same constant. A global section has to be a constant function. The space of global sections is 1-dimensional, spanned by a constant section.

A Flat Line Bundle (Degree 0, with Monodromy):

Let’s construct a non-trivial flat bundle. We use constant transition functions that are not 1: choose $\displaystyle \lambda_1, \lambda_2 \in \mathbb{C}^\times$ .

The cocycle condition is satisfied because $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$ are constants and commute.

A global section must satisfy: $\displaystyle s_{10}(z - 1) = \lambda_1\, s_{00}(z), \quad s_{01}(z - \tau) = \lambda_2\, s_{00}(z)$

The only holomorphic functions on the torus that satisfy these conditions are the zero function, unless $\displaystyle \lambda_1 = \lambda_2 = 1$ (the trivial bundle). This shows concretely that non-trivial flat bundles have no nonzero global sections.

A Non-Flat, Degree 1 Bundle:

Here the transition functions must depend on position. We use a simplified form that satisfies the cocycle condition.

$\displaystyle g_{00,10}(z) = \exp(2\pi i a), \quad g_{00,01}(z) = \exp(2\pi i (z - b)), \quad a,b \in \mathbb{C}$

his pair of functions satisfies the cocycle condition. The non-constant $\displaystyle g_{00,01}$ means the bundle is non-flat: its “twist” depends on the position $\displaystyle z$ . That is two vertical loops which are homotopic give different holonomies.

This bundle has a single global section (up to scalar), given by a theta function. This theta function has exactly one zero on the torus. The location of this zero is determined by the constant $\displaystyle b$ . This establishes a key fact: the continuous family of degree 1 line bundles on the torus is in one-to-one correspondence with the points of the torus itself, via the position of the zero of their section.

Divisors and the Picard Group

There is a more concrete way to understand line bundles using the concept of divisors.

A divisor on a Riemann surface $\displaystyle X$ encodes the zeros and poles of a function:

$\displaystyle D = \sum_{p \in X} n_p \cdot p, \quad n_p \in \mathbb{Z}, \; n_p \neq 0 \text{ for finitely many } p$

$\displaystyle n_p > 0$ : zero of order $\displaystyle n_p$
$\displaystyle n_p < 0$ : pole of order $\displaystyle |n_p|$

The set $\displaystyle \mathrm{Div}(X)$ is an abelian group under addition.

Example: $\displaystyle \mathbb{P}^1$ (Riemann sphere)
Let $\displaystyle f(z) = \frac{z^3}{z^2 - 1}$ .

Zeros: $\displaystyle z=1$ and $\displaystyle z=-1$ , each of order $\displaystyle 1$ .
Pole: order $\displaystyle 3$ at $\displaystyle z=0$ .
At infinity: substituting $\displaystyle z = 1/w$ gives $\displaystyle f(w^{-1}) = w(1-w^2)$ , which has a simple zero at $\displaystyle w=0$ $\displaystyle \Rightarrow$ simple zero at $\displaystyle \infty$ .

Thus: $\displaystyle \mathrm{div}(f) = (1) + (-1) - 3\cdot (0) + (\infty)$

Divisor of product of two rational functions is a sum of the divisors. So we see that the divisors of the of rational functions form the subgroup of principal divisors $\displaystyle \mathrm{Princ}(X)$ .

The Picard Group and Divisors:

The key insight is that any divisor $\displaystyle D$ determines a line bundle $\displaystyle \mathcal{O}(D)$ . Its sections are holomorphic functions permitted to have zeros and poles as prescribed by $\displaystyle D$ : a function $\displaystyle f$ is a section of $\displaystyle \mathcal{O}(D)$ if $\displaystyle \mathrm{div}(f) + D$ is a nonnegative divisor (i.e., defines a global holomorphic function).

The most direct way to build a line bundle $\displaystyle \mathcal{O}(D)$ from a divisor $\displaystyle D = \sum n_p \cdot p$ is to use transition functions defined by ratios of local functions. This provides a clear link between the abstract divisor and the concrete gluing data.

Choose an open cover $\displaystyle {U_i}$ of your Riemann surface $\displaystyle X$ .
On each $\displaystyle U_i$ , find a local rational function $\displaystyle f_i$ whose divisor on $\displaystyle U_i$ is exactly $\displaystyle D_i = D \cap U_i$ .
On the overlap $\displaystyle U_i \cap U_j$ , define the transition function by:

$\displaystyle g_{ij} = \frac{f_j}{f_i}$

This ratio is holomorphic and nonzero on the overlap, which is exactly what a transition function should be.

Sections of $\displaystyle \mathcal{O}(D)$ are defined by local functions $\displaystyle s_i$ on $\displaystyle U_i$ such that

$\displaystyle s_i = g_{ij} \, s_j$

on overlaps. This is equivalent to saying that $\displaystyle s_i f_i$ equals $\displaystyle s_j f_j$ on overlaps. Therefore, $\displaystyle s_i f_i$ defines a single, globally well-defined holomorphic function on the entire surface. Thus, a section $\displaystyle s$ of $\displaystyle \mathcal{O}(D)$ is a rational function whose zeros and poles exactly cancel those of the divisor $\displaystyle D$ .

Let’s use this method to build some line bundles on $\displaystyle \mathbb{P}^1$ .

Example 1: The Trivial Bundle, $\displaystyle \mathcal{O}(0)$

Divisor: $\displaystyle D = 0$ , the zero divisor.
Open Cover: $\displaystyle U_0 = \mathbb{C}$ (coordinate $\displaystyle z$ ) and $\displaystyle U_1 = \mathbb{C}$ (coordinate $\displaystyle w = 1/z$ )
Local Functions: The divisor is zero everywhere, so choose local functions with no zeros or poles: $\displaystyle f_0(z) = 1, \quad f_1(w) = 1$
Transition Function: $\displaystyle g_{01}(z) = \frac{f_1(1/z)}{f_0(z)} = \frac{1}{1} = 1$
Sections: A section $\displaystyle (s_0, s_1)$ must satisfy $\displaystyle s_0 = 1 \cdot s_1$ . Since $\displaystyle s_0$ is holomorphic on $\displaystyle \mathbb{C}$ and $\displaystyle s_1$ on $\displaystyle \mathbb{C}$ , the only global sections are constants. The space of global sections is 1-dimensional.

Example 2: The Hyperplane Bundle, $\displaystyle \mathcal{O}(1)$

Divisor: $\displaystyle D = 1 \cdot (\infty)$ — a single zero at infinity. Degree $\displaystyle 1$ .
Open Cover: $\displaystyle U_0 = \mathbb{C}$ (coordinate $\displaystyle z$ ) and $\displaystyle U_1 = \mathbb{C}$ (coordinate $\displaystyle w = 1/z$ ).
Local Functions: On $\displaystyle U_0$ : $\displaystyle f_0(z) = 1$ (no points in $\displaystyle D$ ).
On $\displaystyle U_1$ : zero at $\displaystyle w = 0$ , so $\displaystyle f_1(w) = w$ .
Transition Function: $\displaystyle g_{01}(z) = \frac{f_1(1/z)}{f_0(z)} = \frac{1/z}{1} = z$ .
Sections: A section $\displaystyle (s_0, s_1)$ satisfies $\displaystyle s_0(z) = z \cdot s_1(1/z)$ .

If $\displaystyle s_1(w) = 1$ : $\displaystyle s_0(z) = z$ .
If $\displaystyle s_1(w) = w$ : $\displaystyle s_0(z) = z \cdot (1/z) = 1$ .

Thus $\displaystyle \mathcal{O}(1)$ has two independent sections: $\displaystyle 1$ and $\displaystyle z$ .

Example 3: The Tautological Bundle, $\displaystyle \mathcal{O}(-1)$

Divisor: $\displaystyle D = -1 \cdot (\infty)$ — a single pole at infinity. Degree $\displaystyle -1$ .
Open Cover: $\displaystyle U_0 = \mathbb{C}$ (coordinate $\displaystyle z$ ), $\displaystyle U_1 = \mathbb{C}$ (coordinate $\displaystyle w = 1/z$ )
Local Functions: On $\displaystyle U_0$ : $\displaystyle f_0(z) = 1$ . On $\displaystyle U_1$ : pole at $\displaystyle w = 0$ , so $\displaystyle f_1(w) = w^{-1}$ .
Transition Function: $\displaystyle g_{01}(z) = \frac{f_1(1/z)}{f_0(z)} = \frac{(1/z)^{-1}}{1} = z$
Sections: A section $\displaystyle (s_0, s_1)$ must satisfy $\displaystyle s_0(z) = z^{-1} s_1(1/z)$ . For $\displaystyle s_0(z)$ holomorphic on $\displaystyle \mathbb{C}$ , $\displaystyle s_1(1/z)$ must vanish at $\displaystyle z=0$ fast enough to cancel $\displaystyle 1/z$ . This forces $\displaystyle s_0 \equiv s_1 \equiv 0$ . Thus $\displaystyle \mathcal{O}(-1)$ has no nonzero global sections.

Equivalence of Divisor and Line Bundles

Two divisors $\displaystyle D_1$ and $\displaystyle D_2$ determine the same line bundle if and only if they differ by a principal divisor—that is, $\displaystyle D_1 - D_2 = \mathrm{div}(f) \quad \text{for some rational function } f.$

So we have the fundamental isomorphism :

$\displaystyle \mathrm{Pic}(X) \;\cong\; \mathrm{Div}(X) / \mathrm{Princ}(X)$

where $\displaystyle \mathrm{Princ}(X)$ denotes the subgroup of principal divisors, i.e., divisors of the form $\displaystyle \mathrm{div}(f)$ for some $\displaystyle f \in \mathbb{C}(X)^\times$ .

We can see this from the fact that the factors $\displaystyle g_{ij} = \frac{f_j}{f_i}$
will be trivial if the divisor is a principal divisor coming from a global function, the local pictures coincide.

The Jacobian Variety and Degree

The degree of a divisor $\displaystyle D = \sum_p n_p P$ is defined by

$\displaystyle \deg(D) = \sum_p n_p.$

For a principal divisor $\displaystyle \mathrm{div}(f)$ , the degree is always zero—this reflects the fact that any rational function has as many zeros (counted with multiplicity) as poles.

Thus, degree descends to a well-defined homomorphism on the Picard group:

$\displaystyle \deg : \mathrm{Pic}(X) \longrightarrow \mathbb{Z}$

The Jacobian variety $\displaystyle J(X)$ is defined as the subgroup of all degree-zero line bundles: $\displaystyle J(X) = \mathrm{Pic}^0(X).$

On an elliptic curve $\displaystyle E$ , we have the remarkable identification:

$\displaystyle \mathrm{Pic}^0(E) \;\cong\; E$

Every degree-zero line bundle is uniquely determined by a point on $\displaystyle E$ . Geometrically, this means that a divisor of the form $\displaystyle (p) - (p_0)$ corresponds directly to the point $\displaystyle p$ on the curve. This is far more concrete than describing the bundle via a pair of constants $\displaystyle (\lambda_1,\lambda_2)$ .

The Jacobian of a Compact Curve

For a compact Riemann surface $\displaystyle X$ of genus $\displaystyle g \geq 1$ , the group of degree-zero line bundles $\displaystyle \mathrm{Pic}^0(X)$ has a special structure: it is a $\displaystyle g$ -dimensional complex torus,

$\displaystyle \mathrm{Pic}^0(X) \;\cong\; \mathbb{C}^g / \Lambda$

where $\displaystyle \Lambda$ is a lattice in $\displaystyle \mathbb{C}^g$ . This complex torus is the Jacobian variety $\displaystyle J(X)$ .

Origin of $\displaystyle \mathbb{C}^g$ : The vector space $\displaystyle \mathbb{C}^g$ arises from the space of global holomorphic 1-forms on $\displaystyle X$ , which has dimension $\displaystyle g$ by Riemann–Roch.

Origin of the lattice $\displaystyle \Lambda$ : The fundamental group $\displaystyle \pi_1(X)$ has $\displaystyle 2g$ generators. Integrating each of the $\displaystyle g$ holomorphic 1-forms along these $\displaystyle 2g$ cycles produces $\displaystyle 2g$ vectors in $\displaystyle \mathbb{C}^g$ , whose integer linear combinations form the lattice $\displaystyle \Lambda$ .

The Jacobian variety $\displaystyle J(X)$ unites analytic information (holomorphic $1$-forms) with topological information (cycles in $\displaystyle \pi_1(X)$ ). The Abel–Jacobi map gives an explicit isomorphism between a degree-zero divisor on $\displaystyle X$ and a point of $\displaystyle J(X)$ , bridging the divisor theory of the curve with the geometry of its Jacobian.

If $\displaystyle D = \sum_{i=1}^n p_i - \sum_{i=1}^n q_i$ is a divisor of degree zero and $\displaystyle {\omega_1, \dots, \omega_g}$ is a basis of the space of holomorphic 1-forms on $\displaystyle X$ , the Abel–Jacobi map is

$\displaystyle \left( \sum_{i=1}^n \int_{q_0}^{p_i} \omega_1 \;-\; \sum_{i=1}^n \int_{q_0}^{q_i} \omega_1, \; \dots, \; \sum_{i=1}^n \int_{q_0}^{p_i} \omega_g \;-\; \sum_{i=1}^n \int_{q_0}^{q_i} \omega_g \right) \; \bmod \Lambda$

where $\displaystyle q_0$ is a fixed basepoint and $\displaystyle \Lambda$ is the period lattice obtained by integrating the $\displaystyle \omega_j$ over a basis of $\displaystyle H_1(X,\mathbb{Z})$ .

This sends $\displaystyle D$ to a well-defined element of $\displaystyle \mathbb{C}^g / \Lambda$ , which is exactly $\displaystyle J(X)$ .

The Generalized Jacobian of a Punctured Curve

When we remove points (punctures) from a curve, its topology changes dramatically: the fundamental group grows more complicated, and the group of degree-zero line bundles $\displaystyle \mathrm{Pic}^0(X)$ also changes. It is no longer a compact torus; instead, it becomes a generalized Jacobian. Formally:

$\displaystyle \mathrm{Generalized\ Jacobian} \ \cong\ J(X_{\mathrm{compact}}) \times G$

where: $\displaystyle J(X_{\mathrm{compact}})$ is the Jacobian of the compactification of $\displaystyle X$ , $\displaystyle G$ is an affine algebraic group (often a product of copies of $\displaystyle \mathbb{C}^\times$ or $\displaystyle \mathbb{C}$ ). The dimension of $\displaystyle G$ depends on the number of punctures and the type of monodromy allowed around them. $\displaystyle J(X_{\mathrm{compact}})$ retains the smooth, topological information of the original curve, while $\displaystyle G$ encodes the new information from punctures—specifically, the monodromy around the new non-contractible loops encircling them.

On a compact torus $\displaystyle T^2$ , a flat line bundle is determined by its holonomies $\displaystyle (\lambda_a,\lambda_b)$ around the two fundamental loops $\displaystyle a$ and $\displaystyle b$ . These holonomies are linked by the complex structure, so the moduli space of such bundles is a $\displaystyle 1$ -dimensional complex torus — the Jacobian $\displaystyle J(T^2)$ .

Adding a puncture at $\displaystyle p_0$ introduces a third non-contractible loop $\displaystyle \gamma_p$ encircling the hole. This loop is independent of the original $\displaystyle a$ and $\displaystyle b$ loops.

Flat connections can now have a simple pole at $\displaystyle p_0$ :

$\displaystyle \omega = \alpha\, dz + \frac{\beta}{z - p_0} \, dz, \quad \alpha, \beta \in \mathbb{C}$

$\displaystyle \alpha$ controls the smooth part — holonomies around $\displaystyle a$ and $\displaystyle b$ — parameterized by $\displaystyle J(T^2)$ . $\displaystyle \beta$ is the residue at $\displaystyle p_0$ , giving monodromy

$\displaystyle \lambda_p = e^{2\pi i \beta} \in \mathbb{C}^\times$

around $\displaystyle \gamma_p$ .

Generalized Jacobian: Since $\displaystyle \alpha$ and $\displaystyle \beta$ are independent, the moduli space is

$\displaystyle J(T^2, p_0) \ \cong\ J(T^2) \times \mathbb{C}^\times$

a $\displaystyle 2$ -dimensional complex manifold:

First factor parameterizes the background connection. ( $\displaystyle J(T^2)$ ). Second factor parameterizes the new monodromy around the puncture. $\displaystyle \mathbb{C}^\times$ .

The Correspondence: Hecke Eigensheaves

With the geometric stage set and the two main players introduced, we can now state the main result: the rank-1 geometric Langlands correspondence. The correspondence is realized through the action of a family of geometric operators, known as Hecke operators, and the central objects are their “eigenvectors,” the Hecke eigensheaves.

Hecke Operators:

For each point $\displaystyle x$ on our Riemann surface $\displaystyle X$ , there is a corresponding Hecke operator, $\displaystyle T_x$ . In the general, non-abelian setting, these are complex integral transforms. However, for the rank-1 case ( $\displaystyle G = \mathrm{GL}(1)$ ), their action is remarkably simple and geometric.

The Hecke operator $\displaystyle T_x$ acts on the Picard group $\displaystyle \mathrm{Pic}(X)$ . It transforms a line bundle $\displaystyle L$ into a new line bundle $\displaystyle L(x)$ , defined as the tensor product of $\displaystyle L$ with the line bundle $\displaystyle \mathcal{O}(x)$ associated to the divisor consisting of the single point $\displaystyle x.$

$\displaystyle T_x : L \ \mapsto \ L(x) := L \otimes \mathcal{O}(x)$

In the language of divisors, if $\displaystyle L$ corresponds to the divisor class $\displaystyle [D]$ , then $\displaystyle L(x)$ corresponds to the divisor class $\displaystyle [D + (x)]$ . The Hecke operator at $\displaystyle x$ simply modifies a line bundle by adding the point $\displaystyle x$ to its divisor class. This operation shifts the degree of the line bundle by one: $\displaystyle \deg(L(x)) = \deg(L) + 1$

Rank-1 Geometric Langlands Correspondence

The rank-1 geometric Langlands correspondence gives a canonical one-to-one relationship between objects on the spectral side and special objects on the automorphic side.

For a Riemann surface $\displaystyle X$ , there is a canonical bijection between

Spectral side: The set of rank-1 local systems on $\displaystyle X$ , i.e. the space $\displaystyle \mathrm{Loc}_1(X)$ .
Automorphic side: The set of rank-1 Hecke eigensheaves on $\displaystyle \mathrm{Pic}(X)$ .

A Hecke eigensheaf is a sheaf on the Picard variety that transforms in a predictable way under the action of all Hecke operators. The “eigenvalues” are determined by the corresponding local system.

The defining property of a Hecke eigensheaf $\displaystyle \mathcal{F}_L$ associated to a local system $\displaystyle L$ is the eigensheaf condition, a geometric analogue of the eigenvalue equation $\displaystyle Av = \lambda v$ :

$\displaystyle T_x(\mathcal{F}_L) \ \cong \ L_x \ \otimes \ \mathcal{F}_L$

$\displaystyle T_x(\mathcal{F}_L)$ is the action of the Hecke operator at $\displaystyle x$ on the sheaf.
$\displaystyle L_x$ is the fiber of the rank-1 local system $\displaystyle L$ at $\displaystyle x$ , a one-dimensional complex vector space.

The monodromy of $\displaystyle L$ around a small loop based at $\displaystyle x$ gives a scalar $\displaystyle \lambda_x \in \mathbb{C}^\times$ —the “eigenvalue” at $\displaystyle x$ .

The local system $\displaystyle L$ provides a complete set of eigenvalues $\displaystyle {\lambda_x}_{x \in X}$ , one for each point of $\displaystyle X$ .
The corresponding Hecke eigensheaf $\displaystyle \mathcal{F}L$ is the joint eigenvector for the entire commuting family of Hecke operators $\displaystyle {T_x}{x \in X}$ .

The problem of classifying local systems is thus transformed into a problem of finding joint eigenvectors for a vast family of commuting geometric operators.

Explicit Construction of the Eigensheaf

For the rank-1 case, the correspondence is not merely an abstract existence result—the Hecke eigensheaf can be constructed explicitly from the local system. This construction, developed by Laumon, Deligne, and others, ties together the key concepts we’ve introduced.

Start with a local system
Begin with a rank-1 local system $\displaystyle L$ on the curve $\displaystyle X$ .
Lift to symmetric products
For each integer $\displaystyle d \ge 0$ , consider the $\displaystyle d$ -th symmetric product $\displaystyle X^{(d)}$ —the space of effective divisors of degree $\displaystyle d$ on $\displaystyle X$ . From $\displaystyle L$ , one constructs a local system $\displaystyle L^{(d)}$ on $\displaystyle X^{(d)}$ .
Descend via Abel–Jacobi
For sufficiently large $\displaystyle d \ (\text{specifically } d \ge 2g-1)$ , the Abel–Jacobi map $\displaystyle \pi_d : X^{(d)} \to \mathrm{Pic}^d(X)$ is a fibration with fibers that are projective spaces. Since projective spaces are simply connected, $\displaystyle L^{(d)}$ is constant along each fiber, hence it descends to a unique local system $\displaystyle \mathcal{F}_{L,d}$ on the degree- $\displaystyle d$ component $\displaystyle \mathrm{Pic}^d(X)$ of the Picard variety.
Extend to all degrees
The Hecke property is then used to define the sheaf on components of lower degree via a reverse-induction process.

The resulting collection of sheaves $\displaystyle \{\mathcal{F}_{L,d}\}_{d \in \mathbb{Z}}$ on the connected components of $\displaystyle \mathrm{Pic}(X)$ forms the Hecke eigensheaf $\displaystyle \mathcal{F}_L$ . By construction—through the Abel–Jacobi descent—it automatically satisfies the eigensheaf condition.

Examples:

An elliptic curve $\displaystyle E$ is modeled as the complex plane $\displaystyle \mathbb{C}$ quotiented by a lattice: $\displaystyle E = \mathbb{C} / (\mathbb{Z} + \tau \mathbb{Z})$

Input Connection:
The connection on $\displaystyle E$ is defined by the 1-form $\displaystyle \eta = 2\pi i \alpha \, dz, \quad \alpha \in \mathbb{C}.$ Monodromies around the $\displaystyle a$ – and $\displaystyle b$ -cycles are: $\displaystyle a = e^{2\pi i \alpha}, \quad b = e^{2\pi i \alpha \tau}$
Lift: For $\displaystyle d$ points $\displaystyle (z_1, \dots, z_d)$ , the lifted connection is: $\displaystyle \eta_{\mathrm{lifted}} = \sum_{j=1}^d 2\pi i \alpha \, dz_j = 2\pi i \alpha \sum_{j=1}^d dz_j$
Descent via Sum Map:
The symmetric product $\displaystyle E^{(d)}$ maps to $\displaystyle E$ by $\displaystyle [z_1,\dots,z_d] \ \mapsto \ u = z_1 + \dots + z_d$ with differential $\displaystyle du = \sum dz_j$ . Thus: $\displaystyle \eta_{\mathrm{lifted}} = 2\pi i \alpha \, du$
Hecke Check:
Adding a point $\displaystyle x$ translates $\displaystyle u \to u + x$ : $\displaystyle (\mathrm{Hecke}_x)^* \eta_{d+1} = 2\pi i \alpha (du_d + dx)$ . Difference: $\displaystyle [2\pi i \alpha (du_d + dx)] - [2\pi i \alpha \, du_d] = 2\pi i \alpha \, dx.$
This equals the input connection $\displaystyle \eta$ evaluated at $\displaystyle x$

Let $\displaystyle X = \mathbb{C} \setminus \{0,1\}$ be the punctured sphere.

Input Connection: $\displaystyle \eta = \frac{\alpha}{z} \, dz + \frac{\beta}{z-1} \, dz, \quad \alpha,\beta \in \mathbb{C}$
Lift: $\displaystyle \eta_{\mathrm{lifted}} = \sum_{j=1}^d \left( \frac{\alpha}{z_j} \, dz_j + \frac{\beta}{z_j - 1} \, dz_j \right)$
Descent via Product Map:
Define: $\displaystyle U_0 = \prod z_j, \quad U_1 = \prod (z_j - 1)$
Using $\displaystyle d\log \left( \prod f_j \right) = \sum d\log(f_j)$ :
$\displaystyle d\log U_0 = \sum \frac{dz_j}{z_j}, \quad d\log U_1 = \sum \frac{dz_j}{z_j - 1}$
Thus: $\displaystyle \eta_{\mathrm{lifted}} = \alpha \, d\log U_0 + \beta \, d\log U_1$
Hecke Check:
Adding $\displaystyle x$ gives $\displaystyle U_0 \to U_0 \cdot x, \quad U_1 \to U_1 \cdot (x-1)$
Difference: $\displaystyle [\alpha \, d\log(U_0 x) + \beta \, d\log(U_1 (x-1))] - [\alpha \, d\log U_0 + \beta \, d\log U_1] = \alpha \, d\log x + \beta \, d\log(x-1)$ is equal
$\displaystyle \frac{\alpha}{x} \, dx + \frac{\beta}{x-1} \, dx$
This is exactly $\displaystyle \eta$ evaluated at $\displaystyle x$ .

Let $\displaystyle X$ be a genus-2 hyperelliptic curve with holomorphic 1-forms $\displaystyle \omega_1, \omega_2$ .

Input Connection: $\displaystyle \eta = 2\pi i (\alpha_1 \, \omega_1 + \alpha_2 \, \omega_2), \quad \alpha_1,\alpha_2 \in \mathbb{C}$
Lift: $\displaystyle \eta_{\mathrm{lifted}} = \sum_{j=1}^d \eta(x_j)$
Descent via Abel Map:
Define Abel–Jacobi coordinates: $\displaystyle u_k = \sum_{j=1}^d \int_{x_0}^{x_j} \omega_k, \quad k=1,2$
Then: $\displaystyle du_k = \sum_{j=1}^d \omega_k(x_j)$
Hence: $\displaystyle \eta_{\mathrm{lifted}} = 2\pi i (\alpha_1 \, du_1 + \alpha_2 \, du_2)$
Hecke Check: Adding a point $\displaystyle x$ gaives $\displaystyle u_k \to u_k + \int_{x_0}^x \omega_k$
Difference: $\displaystyle 2\pi i \left[ \alpha_1 \, d\left( \int_{x_0}^x \omega_1 \right) + \alpha_2 \, d\left( \int_{x_0}^x \omega_2 \right) \right]$
By the Fundamental Theorem of Calculus: $\displaystyle = 2\pi i (\alpha_1 \, \omega_1(x) + \alpha_2 \, \omega_2(x))$
Which is exactly $\displaystyle \eta$ evaluated at $\displaystyle x$ .