[Translate] Automorphic Forms and Geometric Theories

Automorphic Forms and Geometric Theories Robert Langlands

Contents
Contents
 1.  Introduction
   1.1.  Galois Automorphic Group
   1.2.  Remarks on the style and content of this article
 2.  Basic Concepts of the General Theory
 3.  Hypothesis
   3.1.  Case of genus zero
 4.  Reduction Theory for Elliptic Curves
 5.  Correspondences of Hecke Dimensions

1. Introduction Geometric theory of automorphic forms has been introduced by Russian mathematicians, such as Vladimir Drinfeld, and developed by the Russian-American school\footnote{The report “Recent Progress in Geometric Langlands Theory, Bourbaki Seminar, January 2016” is written by Denis Gaitsgory [4], a convenient reference.}. However, I am not satisfied with this theory. The modern arithmetic theory of automorphic forms emerged in the sixteenth century from four sources:

1. the revival of nineteenth-century research;
2. the theory of Hecke;
3. the theory of class fields;
4. and the theory of group representations, in which important names like Frobenius, Weyl, and Chandrasekhar are prominent.

The geometric theory of automorphic forms was introduced by Russian mathematicians, such as Vladimir Drinfeld, and developed by the Russian-American school, but I am dissatisfied with this theory. Modern arithmetic theory of automorphic forms emerged in the 1960s from four sources: the revival of nineteenth-century studies by Siegel, the theory of Hecke, the theory of class fields, and the representation theory of groups, where notable names include Frobenius, Hermann Weyl, and Harish-Chandra. In this arithmetic theory, an indispensable component is the eigenvalues of Hecke operators. Utilizing these numbers, we define extremely important $L$-functions. These numbers are the eigenvalues of Hecke operators, which are defined in arithmetic theory but not in the geometric theory presented by Gaitsgory [4] or Frenkel[3]. The difficulty lies in the fact that in the theory of the Russian-American school, eigen vectors are replaced by eigen bundles, the existence of which is still challenging to prove. Additionally, the description of the classifying space in this theory is given in terms of concepts from bundle theory, stacks, and topological issues introduced to create a classifying space theory that largely satisfies Grothendieck’s functorial requirements. This theory is important, but in my opinion, it is not the theory necessary for statements and proofs of a geometric nature, which I referred to in the arithmetic theory as functorialities and reciprocities.

For reasons I will explain later, it may be better to speak of duality in geometric theory, but functoriality is a consequence of duality. One of the goals of arithmetic theory is to prove functoriality and, using this functoriality, to construct the automorphic Galois group^[1]. In geometric theory, however, this Galois group is already given. Nevertheless, it is necessary to prove that this group possesses the desired properties. What is needed to understand the foundations of the geometric theories introduced in this article are commonly accepted definitions from the realms of sets, spaces, and measures.

The goal of this article is to describe what seems to me a suitable analogue of arithmetic theory and to prove it for an interesting, even striking, yet easily accessible case: the group $\mathrm{GL}(2)$ over an elliptic curve. However, there are two geometric theories, one over a finite field and another, as in this article, over the field of complex numbers. I have not thoroughly examined [4] or [5], but it seems (!) to me that [5]*{Th. 0.1} suggests a theory for the function field over a finite field that is compatible with Andre Weil’s Rosetta Stone^[2]. Moreover, and I find it very interesting, according to [5]*{Th. 0.1}, for such a field, the automorphic Galois group is isomorphic to $\mathrm{Gal}(\overline{F}/F)$. On the other hand, just as the Paley-Wiener theorem or the Fourier transform theory for Schwartz generalized functions does not replace the $L^2$ theory of Fourier transformations on $\mathbb{R}$ or on $\mathbb{R}^n$, $n = 2, 3, \ldots$, in my opinion, the theory proposed in [4] does not replace the $L^2$ theory of Hecke operators. I believe that such a theory exists for every reductive group on an arbitrary Riemann surface. At times, I will touch upon the general case, but I haven’t seriously contemplated it. The goal of this article is to describe the theory for the group $\mathrm{GL}(2)$ over an elliptic curve, for which it is already more accessible. While I can imagine a general spectral view of geometric theory, I cannot imagine a theory that unites it with the one described in [4]. As a final observation, geometric theory relates in part to complex-differential geometry, but this article is written for specialists in the theory of automorphic forms, to whom this geometric theory may be unfamiliar. Due to such inherent weakness, I have included some elementary explanations of concepts in [2]. These explanations often constitute elongated digressions. Moreover, I allowed myself many other digressions, some quite elementary, assuming that most readers, like me, have as little experience with differential geometry as I do.

The conclusion is somewhat abrupt, but the appearance of a decisive sign in a place where I least expected it confirmed my confidence in the correctness of my conviction. I no longer needed to convince anyone of this.

The exposition is not detailed enough, but the topic is new, and our understanding is incomplete, so a complete explanation would be inappropriate here.
1.1. Galois Automorphic Group Since the main goal of this article is an introduction to geometric theory and the preliminary –– possibly temporary, possibly final –– construction of the corresponding mathematical subject, I prefer to define it immediately. This implies the introduction of a group that comes with prescriptions for transforming a homomorphism from it to the ${}^LG$ group into a proper conjugate section. These sections are defined below. These prescriptions can be intricate. For example, they make use of the Atiyah-Bott theorem. In any case, we confirm the reciprocity law for $\mathrm{GL}(2)$ and an elliptic curve, first by computing the eigenvalues of the Hecke operators and then the Yang-Mills connections. Without assuming that this endeavor has the same significance as Dedekind’s achievements, just as a casual remark, I propose another title, “Was ist und was soll die geometrische Theorie der automorphen Formen?”
1.2. Remarks on the style and content of this article It is written for mathematicians familiar with the theory of automorphic forms over number fields but, like me, with limited knowledge of differential geometry, especially vector bundles and connections. Therefore, considerable space is devoted to simple or well-known concepts. A little acquaintance with them is insufficient, as I realized when studying [1] and [2]. The explanations by these authors are concise and smooth but seldom precise and rarely detailed, apparently because they assumed the reader had some familiarity with the fundamental concepts. I thought I had it, but initially, it was not enough for a precise understanding of their deductions and even their fundamental assertions. Therefore, I have added here all the additional reflections that I deemed necessary or useful, but only as needed.

There is a simple but useful remark. In the arguments of this article, there are three degrees: the clear establishment of eigenfunctions and eigenvalues of Hecke operators; the clear establishment of Yang-Mills connections; their comparison.

I came to an understanding of the necessary foundation of the theory only during the reading of references and writing the article, so it started with an assertion for which proof could only be provided slowly as I came to an understanding of the relevant theory. My initial understanding of the proven theorem did not contain either an exact statement or a profound knowledge of the differential-geometric foundation. Therefore, there was much that I could not anticipate and was not prepared for. This led to redundancies and digressions, some elementary, which might be burdensome for a geometer but could possibly help specialists in fields related to the fundamental problem of this article, namely, reciprocity and functoriality in the theory of automorphic forms, not only in one of its three varieties but in all, although this article is dedicated exclusively to geometric theory.

In the end, I did not state the theorem, although I could have, and the reader could infer the corresponding statement. What is striking is the exact coupling of the calculations of the eigenconjugate sections of Hecke and the structure of the Yang-Mills theory. Although at the moment I do not have a precise recollection of the calculations in classical theory (either the initial calculations in Gauss or the later calculations in Hasse, and I am beginning to fear that I have not studied them enough), it seems to me that there is an unrecognized similarity between them and those in this article. In both cases, two seemingly different objects are connected by a third sensitive clock mechanism. In this article, it looks like a sequence of similarities between two quite complex but specific calculations, and in the very end, the comparison can only be completed thanks to the remarkable detail that appears in both, although for different reasons. To hide this in the statement of the theorem, which, in the end, would only be a special case, seemed rude.

The construction of the article is simple. By the end of section Section 3, the goals of functoriality in a geometric context are clarified. In Section 4, I recall the classical theory of elliptic curves, from which we draw to make our discussion as solid as possible. The next three sections are devoted to introducing Hecke operators and calculating their eigenfunctions and eigenvalues, but within the framework of Hilbert space. After that, in sec:VIII and sec:IX, the necessary differential geometry is recalled, sometimes at a foundational level. In sec:X and sec:XI, further discussion of the Yang-Mills theory leads to a comparison with early calculations in the Hecke theory, and then easily to the desired comparison. The last section is brief. I allowed myself many digressions, some quite elementary, assuming that most readers would have as little experience with differential geometry as I do.

Finally, a word about language. When I enrolled in university in Vancouver at the age of sixteen, coming from a village, I was initiated into a new world not only of mathematics but also informed that there are many languages in the world, some of which are currently necessary for a successful life as a mathematician. Although slowly, I learned that they offer much more than just mathematics itself. Unfortunately, for reasons that need not be detailed here, the mathematical profession no longer offers this window to the world. Nevertheless, the desire to write an article in Russian remained. This article was my last opportunity to do so.

Reflecting on this article after it was completed and on its style, I was puzzled by many digressions and their relation, but then the reason became clear. It is related to the fact that the main part of the work is dedicated to my efforts to understand concepts taken from [1] and a small part of [2]. After the structure of $\mathrm{Bun}_G$ and the nature of Yang-Mills connections became clear, my only idea was to introduce Hecke operators as operators in Hilbert space. It was simple, though new and even somewhat revolutionary, as mathematicians’ attachment to bundles was universal. There are no bundles in this article.

In the final conclusion, I emphasize my dissatisfaction with what is presented here, but a sufficient report requires a brief but complete general presentation of the Yang-Mills theory, as well as an understanding of the last paragraphs of this article, where direct images of sheaves and induced representations appear, but understood under general conditions. Currently, this is hardly accessible. This is a task proposed to the reader.

2. Basic Concepts of the General Theory

First and foremost, we need a reductive algebraic group $G$ over a compact non-singular algebraic curve $M$ over $\mathbb{C}$ and its classifying space $\mathrm{Bun}_G$. I will explain a general proposition that I will mainly prove for the group $G = \mathrm{GL}(2)$ over an elliptic curve $M$. This case is already remarkable. I use conclusions from the article by Atiyah[1]. Perhaps they are sufficient also for the group $\mathrm{GL}(n)$.

Let $\mathbb{F}$ be the field of meromorphic functions over $M$. Let $\mathbf{A}_{\mathbb{F}}$ be the adelic algebra of the field $\mathbb{F}$, and $\mathbb{F}_x$ be the local field at the point $x\in M$. The ring $\mathcal{O}_x$ is the ring of integers in $\mathbb{F}_x$, and $\mathcal{O} = \prod_x\mathcal{O}_x$. The quotient
$$
\mathrm{Bun}_G = G(\mathbb{F}) \backslash G(\mathbf{A}_{\mathbb{F}}) / G(\mathcal{O})
$$
is known. It is proven in ([3]*{Sect.~3}). I will explain it below. Later, we will briefly explain how, for us, $\mathrm{Bun}_G$ represents a topological space, although not a Hausdorff space, and that it carries a locally metric structure and measure $\mu$. For $G = \mathrm{GL}(2)$ and an elliptic curve $M$, the structure and measure are simple.

Hecke Operators – linear mappings of the space $L^2(\mu)$. The main theme of this article is Hecke operators and their eigenvalues. For each point $x\in M$, there exists a commutative algebra of Hecke operators $\mathfrak{H}_x$. These algebras are commutative and pairwise commutative. Let $\Theta \in\mathfrak{H}_x$, then its Hermitian conjugate operator $\widetilde\Theta$ is also a Hecke operator. Therefore, there exists a corresponding spectral decomposition of the space $L^2(\mu)$, and the goal of this article is to describe the eigenvalues and eigenfunctions of this decomposition. Although this article predominantly discusses elliptic curves, I cannot refrain from a few general remarks. With God’s help, I will return to the general curve later. In the next section, I will formulate general assumptions, which I will prove later for an elliptic curve. Each Hecke operator $\Theta$ will be defined by a correspondence $\Theta$. This correspondence is a subset of the set $\mathrm{Bun}_G \times \mathrm{Bun}_G$. This correspondence also carries the support of a measure, which will be better described later.

3. Hypothesis While we will prove these assumptions only for curves of the first kind and not for higher genera, and only for the group $\mathrm{GL}(2)$, it seems to me that they are also valid for $\mathrm{GL}(n)$, thanks to the article [1]. In general, there are two steps:
\begin{enumerate*}

replace $\mathrm{GL}(2)$ with another group;

\item replace $M$ with an arbitrary compact Riemann surface.
\end{enumerate*}
I have not thought seriously about these yet. I will propose, however, a compelling hypothesis, but to assert it for $g > 1$, we will need to consider the intricacies of the structure of $\mathrm{Bun}_G$. There is, of course, another step, branched theory, but I have never contemplated that.

It is known that the Hecke algebra of the group $G = \mathrm{GL}(2)$ or any reductive group –– this section deals with this –– is isomorphic to the ring of representations of the dual group ${}^LG$, and that each homomorphism of this algebra into $\mathbb{C}$ is given by a semi-simple class $\theta\in{}^LG$. Consequently, the eigenfunction of all Hecke operators, or better, the eigenvalues correspond to the function whose values at the point $x \in M$ represent the semi-simple class $\left\{\theta(x)\right\}$. We call this an eigenconjugate section or, more briefly, an eigensection. The structure of the set of these sections is unclear. It is known that there is a similar theory for all $M$ and all $G$, and we consider the general case in this section.

In the theory of automorphic forms, the concept of functoriality expresses that the set of sections or the set of those sections belonging to the $L^2$-theory is given by homomorphisms, or unitary homomorphisms, of the assumed automorphic Galois group into ${}^LG$. It seems that the difference between arithmetic and geometric theory lies in the fact that geometric theory can be described by a simple use of ordinary concepts. In arithmetic theory, this is not the case. In this section, I describe a general hypothesis that I will prove later, after some preparation.

The concept related to the automorphic Galois group is given in the article by Atiyah and Bott [2]. This article, together with the aforementioned Atiyah’s article, has had a significant impact on this paper. This concept is the group $\Gamma_{\mathbb{R}$ ([2]*{Thm. 6.7}). For a curve of genus $g$, this group is an extension of the central extension
$$
1 \to \mathbb{Z} \to \Gamma \to \pi_1(M) \to 1, \quad \Gamma_{\mathbb{R}} = \mathbb{R} \times_{\mathbb{Z}} \Gamma.\tag{1.a}
$$
The group $\Gamma$ is generated by elements $A_1,\ldots,A_g$, $B_1,\ldots,B_g$, and $J = 1 \in \mathbb{Z}$ with a single relation,
\[
A_1B_1A_1^{-1}B_1^{-1}A_2B_2A_2^{-1}B_2^{-1} \cdots A_gB_gA_g^{-1}B_g^{-1} = J, \quad J = 1 \in \mathbb{Z}.\tag{1.b}
\]
In the following sections, where $g = 1$, it is established that $A_1, B_1$ represent loops $(0, 2\omega_1)$, $(0, 2\omega_2)$ in the elliptic curve $M$.

We are only interested in representations $\phi$ of the group $\Gamma$ such that the orders of the elements $\phi(A_i)$, $\phi(B_i)$, and $\phi(J)$ are all finite. They are called admissible. We also need the group $\widetilde\Gamma = \mathbb{Z} \times \Gamma$. It is possible that the order of the elements $\phi(z \times 1) \in \mathbb{U} \times 1$, $z \in \mathbb{Z}$, $\mathbb{U} = \left\{ w \in \mathbb{C} | |w| = 1\right\}$, $1 \in\Gamma$, is infinite. The term $\mathbb{Z}$ does not manifest in [2], because in this article, the Chern class of the bundle is chosen so that $\mathrm{Bun}_G$ becomes connected, i.e., the connected component of the correct $\mathrm{Bun}_G$. Embedding this $\mathbb{Z}$ into $\widetilde\Gamma$ is somewhat arbitrary. It is related to the choice of the bundle $A = A_0$ in [1]*{Thm. 6} and Section 4 of this article.
\[
A_1B_1A_1^{-1}B_1^{-1}A_2B_2A_2^{-1}B_2^{-1} \cdots A_gB_gA_g^{-1}B_g^{-1} = 0 \times J.\tag{1.c}
\]

I call this new group the automorphic Galois group. More precisely, the automorphic Galois group $\Gamma_{\mathrm{aut}}$ is the product of $\mathbb{Z}$ and the inverse limit of all finite groups of fractions of the group $\widetilde\Gamma$. In this group (at each finite level of this group – which is what matters), the orders of the images of $\phi(A_i)$, $\phi(B_i)$ and $\phi(J)$ are all finite. For example, first, we build the intersection $\Gamma_2$ of all kernels of homomorphisms from the group $\Gamma$ to a group with an order divisible by $2$, then the intersections $\Gamma_{36}$, $\Gamma_{27000}$, and so on. Then
\[
\Gamma_{\mathrm{aut}} = \lim_{\leftarrow} \mathbb{Z}/n(k)\mathbb{Z} \times \lim_{\leftarrow} \Gamma/\Gamma_{n(k)},\tag{1.d}
\]
where \( n(k) = (k!)^k \). This annoying subtlety is necessary because connectivity and eigenconjugate sections are interrelated but distinct. In particular, the additional $\mathbb{Z}$ is needed because $\mathrm{Bun}_G$ is not connected. As will be explained later, $\Gamma_{\mathbb{R}}$ and $\Gamma_{\mathrm{aut}}$ are both defined as modifications of the group $\Gamma$. Eigenconjugate sections and Yang-Mills connections are closely related. I also note that in $\Gamma_{\mathrm{aut}}$, the equation $J = 1$, $1 \in \mathbb{Z}$ is replaced by
\[
J = 1,\quad 1 \in \lim_{\rightarrow} \mathbb{Z}/n\mathbb{Z},
\]
where \( z \mapsto nz/m \), \( m|n \), in an increasing sequence from left to right.

Let ${}^LG_{\mathrm{unit}}$ be a compact form of the group ${}^LG$. I assume this, but only approximately because this proposition allows avoiding endoscopic difficulties, that there exists a bijective map from the set of eigenconjugate sections to the set of homomorphisms from the group $\Gamma_{\mathrm{aut}}$ to ${}^LG_{\mathrm{unit}}$. The goal of this article is to prove this for the group $\mathrm{GL}(2)$, although it seems to me that the article [1] would allow obtaining a proof for $\mathrm{GL}(n)$. However, there is a difficulty here. I still do not know how to recognize irreducible representation by the group $\Gamma_{\mathrm{aut}}$ in $\mathrm{GL}(n)$ if the size of $n$ is greater than two, even for an elliptic curve. It will be clear to the reader that I began to understand the articles [2] and [1] only when I wrote this article. Initially, my ambition was more modest, but now it seems to me that the main statements of the geometric theory over $\mathbb{C}$ are surprisingly simple, though not obvious. I do not know if readers of [4] or [5] need or even find it useful to understand them.
3.1. Case of genus zero If the genus is zero, equation (1.b) is meaningless, so $\Gamma_{\mathrm{aut}}$ needs to be defined differently. It seems to me that there is no other choice but to assume that it is equal to $\mathbb{Z}$. This is consistent with the fact that for genus zero, all bundles are direct sums of line bundles, each of which is itself a power of a single line bundle of degree zero. This case is explicitly excluded from the discussion in [2]. Throughout this article, it will be clear to the reader that it would have been better written, both in terms of clarity and accuracy, if I had a better understanding of the Yang-Mills theory in particular and differential geometry in general.

There are two important notes. Firstly, I will only consider the untwisted theory. Secondly, in arithmetic theory, the existence of the Galois automorphic group is equivalent to functoriality ^[3]. The close connection between the theory of algebraic numbers and the theory of algebraic curves over $\mathbb{C}$ was described by Dedekind and Weber in the article ^[4]. The theory of algebraic curves over the Galois field was added by André Weil ^[5].

The theories of the two papers [1] and [2] are mostly unfamiliar to me and my readers. Therefore, I will extensively explain a few concepts.
4. Reduction Theory for Elliptic Curves This important theory has reached its final form over the field of algebraic numbers in the report by Borel and Harish-Chandra [1], but it has not yet been fully developed for Riemann surfaces. In the article [1], Lemma 4 marks the beginning of this theory for arbitrary genus, but for $g = 1$, the completed theory exists for $GL(2)$ and even for $GL(n)$ in the article itself, which I primarily explain for $GL(2)$ but without proofs. This theory is a complete description of BunG, that is, a description of all two-dimensional vector bundles. To prove it, one needs modern bundle theory, but to describe it, I prefer to apply the Weierstrass theory as presented in the book by Whitaker and Watson [1]. Of course, this is not necessary. Rather, it is a test of my knowledge of the theories of Atiyah and Atiyah-Bott, and an indication of my mathematical inclinations!

The concepts in [2] that reflect modern notions of complex differential geometry, and therefore topology and complex analysis, are exceptionally elegant. However, they are abstract, and it’s easy for a novice to overlook their complexity and subtlety, which often happened to me when reading their works. I have come to the conclusion that a concrete expression of them guards against misunderstandings. So, if $L = 2\mathbb{Z}\omega_1 \oplus 2\mathbb{Z}\omega_2$, $\omega_1, \omega_2 \in \mathbb{C}$, and $\omega_1/\omega_2 \notin \mathbb{R}$, then the curve $M = \mathbb{C}/L$. In this article, an $GL(n)$-bundle is a matrix-valued meromorphic function $M(z), z \in \mathbb{C}$, such that $M(z + \lambda) = M(z)K_\lambda(z)$ for all $\lambda \in L$, where the matrix $K_\lambda$ is holomorphic.

In the theory of Weierstrass, the sigma function $\sigma(z), z \in \mathbb{C}$, plays a fundamental role. Here are its properties: (i) it is holomorphic; (ii) its expansion in a power series $\sigma(z) = z + \ldots$; (iii) $\sigma(z + 2\omega_1) = -e^{2\eta_1(z+\omega_1)}\sigma(z)$, $\sigma(z + 2\omega_2) = -e^{2\eta_2(z+\omega_2)}\sigma(z)$; (iv) $\eta_1\omega_2 – \eta_2\omega_1 = \pi i/2$; (v) if $\sigma(z) = 0$, then $z \in L$.

Let $a_1, \ldots, a_m, b_1, \ldots, b_n$ be points in $\mathbb{C}$. Then
\begin{equation}
\phi(z) = \frac{\sigma(z-a_1)\sigma(z-a_2)\ldots\sigma(z-a_m)}{\sigma(z – b_1)\sigma(z – b_2) \ldots \sigma(z – b_n)}\tag{2}
\end{equation}
is a meromorphic function of $z\in \mathbb{C}$. Moreover
\begin{align*}
\phi(z + 2\omega_1) &= \phi(z)(-1)^{m-n}e^{2\eta_1(m-n)(z+\omega_1)}e^{-2\eta_1\left(\sum_{i=1}^{m}a_i-\sum_{j=1}^{n}b_j\right)}\\
&= \phi(z)(-1)^{m-n}e^{2\eta_1(m-n)(z+\omega_1)}e^{-2\eta_1\theta}\\
\phi(z + 2\omega_2) &= \phi(z)(-1)^{m-n}e^{2\eta_2(m-n)(z+\omega_2)}e^{-2\eta_2\left(\sum_{i=1}^{m}a_i-\sum_{j=1}^{n}b_j\right)}\\
&= \phi(z)(-1)^{m-n}e^{2\eta_2(m-n)(z+\omega_2)}e^{-2\eta_2\theta}\\
\end{align*}
where \( \theta = \sum_{i=1}^{m}a_i – \sum_{j=1}^{n}b_j \). The function \( \phi \) is periodic only if \( m = n \). In this case, it is an ordinary \( \phi \) function over \( \mathbb{C} \) covering the curve \( M \). If \( m = n \) and \( \theta = 0 \), then \( \phi \) is a function over \( M \). If \( \sigma \) is replaced by the function \( \sigma'(z) = \sigma(z)\exp(\lambda z) \), \( \lambda \in \mathbb{C} \), then equations (iii) are replaced by
\[
\sigma'(z + 2\omega_i) = -e^{2\eta_i+2\lambda\omega_i} \sigma'(z) = -e^{2\eta_i’} \sigma'(z), \quad \eta_i’ = \eta_i + \lambda\omega_i
\]
That is, these equations are just a normalization of the function \( \sigma \), or if you like, \( \sigma \) and \( \sigma’ \) define different but equivalent linear sheaves. Equation (iv) remains unchanged. Equations (iii) define gluing and do not change the sheaves. In the region \( 2x\omega_1 + 2y\omega_2 \), \( 0 \leq x, y < 1 \), the function \( \sigma \) has only one zero. This is the reason for equation (iv). (Here note that: The correct condition for the divisor \( (a_1, \ldots, a_n) \) to be equivalent to \( (b_1, \ldots, b_n) \) is described by the equation \[ \theta = \sum_{i=1}^{n}a_i - \sum_{j=1}^{n}b_j \in 2\mathbb{Z}\omega_1 + 2\mathbb{Z}\omega_2, \quad k=1,2. \] That is, if these equations hold, there exists \( \lambda \in \mathbb{C} \) such that the function \( \exp(-\lambda z)\phi(z) \) is periodic with respect to \( 2\mathbb{Z}\omega_1 + 2\mathbb{Z}\omega_2 \). For this to happen, it is necessary that \( -2\lambda\omega_k - 2\eta_k\theta \in 2\pi i\mathbb{Z} \) for \( k = 1, 2 \). Let \( \theta = 2\omega_1 \). There are two numbers: for \( k = 1 \), \( -2\lambda\omega_1 - 4\eta_1\omega_1 \), and for \( k = 2 \), \( -2\lambda\omega_2 - 4\eta_2\omega_1 \). If \( \lambda = -2\eta_1 \), then the first number is \( 0 \) and the second number is \( 4(\eta_1\omega_2 - 4\eta_1\omega_2) = 2\pi i \). If \( \theta = 2\omega_2 \), the argument is similar. Such considerations are redundant but comforting.) In general, \( \phi \) defines a linear sheaf \( \Lambda = \Lambda(a_1, \ldots, a_m, b_1, \ldots, b_n) \) for which it itself is a multi-valued section. Two sheaves \( \Lambda \) and \( \Lambda'=\Lambda(a'_1, \ldots, a'_{m'}, b'_1, \ldots, b'_{n'}) \) are isomorphic if and only if \( m - n = m' - n' \), and \[ \sum_{i=1}^{m}a_i-\sum_{j=1}^{n}b_j = \sum_{i=1}^{m'}a'_i-\sum_{j=1}^{n'}b'_j \mod 2\pi\omega_1\mathbb{Z}+2\pi\omega_2\mathbb{Z} \] Then the degree of the sheaf is \( n - m \). In my opinion, this description of linear sheaves is the clearest, but for sheaves of higher dimension, like those in Atiyah's article, cohomological methods are often needed. Atiyah's theorems and lemmas are detailed, possibly because the sets he deals with are stacks, although there are no stacks in Atiyah's article or here. For him, they are spaces, and for us, they are spaces with a local metric and measure. Here, I want to primarily describe the space \( \mathrm{Bun}_{GL(2)} \) by imitating Atiyah. Above the elliptic curve, there are two types of two-dimensional sheaves, decomposable sheaves \( \Phi = \Lambda_1 \oplus \Lambda_2 \), and Atiyah-type sheaves, i.e., others. Let \( \mathfrak{D}(m, n) \) be the set of \( \Phi = \Lambda_1 \oplus \Lambda_2 \) for which \( \text{deg} \, \Lambda_1 \), i.e., the degree of \( \Lambda_1 \), equals \( m \) and the degree of \( \Lambda_2 \) equals \( n \). The set of Atiyah-type sheaves is the union \[ \left\{ \bigcup_{m \in \mathbb{Z}} \mathfrak{A}(m,m) \right\} \cup \left\{ \bigcup_{m \in \mathbb{Z}} \mathfrak{A}(m+1,m) \right\} \] For general curves, there are decomposable and indecomposable sheaves. In Atiyah's article, the set of these latter ones is denoted by \( E(r, d) \), where \( r \) is the rank and \( d \) is the degree. Before I describe these sheaves, I note that \( \Lambda_1 \oplus \Lambda_2 \) is equivalent to \( \Lambda'_1 \oplus \Lambda'_2 \) if and only if \( \{\Lambda_1, \Lambda_2\} = \{\Lambda'_1, \Lambda'_2\} \). Therefore, \( D(m, n) \) is a two-dimensional complex manifold. If \( m = n \), then it is a singular curve for which \( \Lambda_1 = \Lambda_2 \). In contrast to arithmetic theory, in geometric theory everything is precise. That is, the fundamental domain is precisely described. In Atiyah's article (Lemma 3), as the first step and as a consequence of the Riemann-Roch theorem for sheaves of higher dimensions, it is proven that for a two-dimensional sheaf over an elliptic curve, there exists a representative \[ \Theta = \begin{pmatrix} \Lambda_1&*\\ 0 &\Lambda_2 \end{pmatrix},\quad \mathrm{deg}\Lambda_2\leq 2+m\mathrm{deg}\Lambda_1. \] Although this is not necessary, Atiyah prefers to assume the sufficiency of sections for this sheaf \( \Theta \), i.e., that for each point \( x \in M \), the map \( \Gamma(\Theta) \rightarrow \Theta_x \) is surjective, where \( \Gamma(\Theta) \) consists of sections of \( \Theta \). To achieve this, it is sufficient to replace the sheaf \( \Theta \) with the sheaf \( \Theta' = \Lambda \otimes \Theta \), where \( \Lambda \) is a suitable linear sheaf. Each conclusion for \( \Theta' \) is also a conclusion for \( \Theta \). In general, if a sheaf of any dimension has sufficient sections, then it has an upper triangular representative. As the second lemma (Lemma 6'), Atiyah asserts that if the sheaf \( \Theta \) is indecomposable and if \( \Gamma(\Theta) \neq 0 \), then it has such a maximal splitting \[ \Theta\cong \begin{pmatrix} \Lambda_1 & * & * & \cdots & * \\ 0 & \Lambda_2 & * & \cdots & * \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \Lambda_n \end{pmatrix} \] where \( \Lambda_i \geq \Lambda_1 \geq 1 \), \( \Gamma(\text{Hom}(\Lambda_1, \Lambda_i)) \neq 0 \) for \( i = 2, \ldots, n \), and \( \Gamma(\Lambda_1) \neq 0 \). At this moment, \( n = 2 \), but it is useful to speak about the general case. This is again a consequence of the Riemann-Roch theorem. However, Atiyah's following argumentation is difficult, and I want to explain only the conclusions that are important for us, sometimes postponing explanation, and others that we do not need, even skipping them. I use old-fashioned concepts. In the theory of Weierstrass, there is a second important function, this function \[ \zeta(z) = \frac{d \ln \sigma(z)}{dz} = \frac{\sigma'(z)}{\sigma(z)} \] It satisfies additive conditions, \[ \zeta(z+2\omega_1) = \zeta(z) + 2\eta_1, \quad \zeta(z+2\omega_2) = \zeta(z) + 2\eta_2. \] In the fundamental domain for the group \( L \subset \mathbb{C} \), it has only one pole, located at the point \( 0 \). The matrix \[ M(z) = \begin{pmatrix} 1 & \zeta(z) \\ 0 & 1 \end{pmatrix} = \exp \begin{pmatrix} 0 & 0 \\ \zeta(z) & 0 \end{pmatrix} \tag{5} \] satisfies the multiplicative conditions, \[ M(z+2\omega_i) = M(z) \begin{pmatrix} 1 & 2\eta_i \\ 0 & 1 \end{pmatrix}, \quad i=1,2. \] Therefore, it defines a \( GL(2) \)-sheaf \( \Pi \). If \( \Lambda \) is a linear sheaf, then \( \Lambda \otimes \Pi \) is also a \( GL(2) \)-sheaf. \( \Lambda \otimes \Pi \) and \( \Lambda' \otimes \Pi \) are equivalent only if \( \Lambda \) and \( \Lambda' \) are equivalent ([1]*{Th. 5}). The degree \( \text{deg}(\Lambda \otimes \Pi) \) equals \( 2 \text{deg} \, \Lambda \). The set of such sheaves with degree \( 2m \) is the set \( \mathfrak{A}(m, m) \). A more general form of definition (4) is given in ([1]*{Th. 5}) \[ F_r = \exp \begin{pmatrix} 0 & \zeta(z) & 0 & \cdots & 0 \\ 0 & 0 & \zeta(z) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \zeta(z) \\ 0 & 0 & 0 & \cdots & 0 \\ \end{pmatrix}\tag{6} \] a matrix of order \( r \). It will be necessary to prove that \( F_r \) is indecomposable, but we confine ourselves to the group \( GL(2) \). The location of the pole can be shifted by multiplying on the left by \( \begin{pmatrix} 1 & h(\cdot) \\ 0 & 1 \end{pmatrix} \), where \( h(\cdot) \) is a meromorphic function with poles at 0 and any other point for which, of course, the sum of the residues equals zero. Defining the set \( \mathfrak{A}(m + 1, m) \) in one sense is more difficult because there is no canonical choice, but in another sense, it is easier. It consists of \[ F(d) = \exp \begin{pmatrix} 1 & \phi(z) \\ 0 & \sigma^{-1}(z-d) \end{pmatrix}, \quad \phi(z) = \frac{\sigma(z-a_1)\sigma(z-a_2)}{\sigma(z-b_1)\sigma(z-b_2)}, \tag{7}\] where \( a_1, a_2,b_1,b_2 \) are fixed, \( a_1 + a_2 = b_1 + b_2 \), \( a_1 - a_2 \neq b_1 - b_2 \), but \( d \) is variable. This means that as a set \( A(m, m) \simeq A(m, m + 1) \simeq M \). It may seem somewhat arbitrary, but one needs to look through [1] to understand that it is inevitable. In passing, I admit that understanding [1] is difficult for me. Describing Atiyah's general conclusions (\cite[Th. 6]{Atiyah1957Vector}) is useful, and we need it in this article. He describes all indecomposable sheaves \( M \) of dimension or rank \( r \) and degree \( d \). It is useful to first select a given linear sheaf \( \Lambda_0 = \Lambda_{A_0} \) of degree one. This sheaf is determined by the chosen point \( A_0 \). Then the sheaf of each degree \( d \) is defined: if \( d = 0 \), the sheaf is trivial; if \( d > 0 \), there is a section with a single pole and its degree is \( d \) at the point \( A_0 \), but no zeros; if \( d < 0 \), the zero is replaced by a pole. With these choices, for a given \( r \), the sets \( \mathcal{E}(r, d) \), which we will introduce, are all identified. Now let \( d = ar + d' \), where \( 0 \leq d' < r \). If \( a = 0 \), we move directly to the second stage. If \( a > 0 \), then

\[ N = A^a \otimes N’ \]

where \( N’ \) is an indecomposable sheaf of dimension \( r’ = r \) and degree \( d’ \). Therefore, we can move to the second stage and assume that \( d’ = d-ar < r' = r \). If \( d' = 0 \), this stage is final, but if \( d' > 0 \),
\[ N’=\begin{pmatrix} I & *\\0&N\prime\prime \end{pmatrix}, \]
where \( I \) is the identity matrix of rank \( s < r' \). Let \( r'' \) be the rank of \( N'' \). Then \( r'' \) is less than \( r' \), and the degree \( d'' = d' \). The exact form of the matrix \( * \) is irrelevant. What matters is only that \( M' \) is indecomposable. Then the rank \( r'' = r - s < r \). The initial doubling appears here again. Continuing, we will arrive at the pair \( (r', d' = 0) \), \( N' = F_{r'} \). For sheaves of higher order, we can replace (7) with the matrix \[ \begin{pmatrix} 1 & 0 & \cdots & 0 & \phi(z) \\ 0 & 1 & \cdots & 0 & \phi(z) \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & \phi(z) \\ 0 & 0 & \cdots & 0 & \sigma^{-1}(z-d) \end{pmatrix} \] The set \( \text{BunGL}(r) \) is a construction in which there are \( r \) steps between floors. Atiyah's description of this transformation is different but very instructive. It is useful to provide it with some additional details. Let \( \mathcal{E}(r, d) \) be the set of indecomposable sheaves of dimension \( r \) and degree \( d \), and let \( h \) be the greatest common divisor of \( r \) and \( d \). Atiyah describes the mapping \( \alpha_{r, d} : \mathcal{E}(h, 0) \rightarrow \mathcal{E}(r, d) \) which is inverse to our transformation. That is, he constructs elements in \( \mathcal{E}(r, d) \) while we analyze them. The composite parts of his construction are: (i) \( \alpha_{r, 0} : F_r \rightarrow F_r \); (ii) \( \alpha_{r, d+r}(E) : E \rightarrow A \otimes \alpha_{r, d}(E) \); (iii) if \( 0 < d < r \), then \begin{equation} \alpha_{r, d}(E)=\left(\begin{array}{ccccccccc} 1 & 0 & \cdots & 0 & * & \cdots & & \cdots & * \\ 0 & 1 & \cdots & 0 & * & \cdots & & \cdots & * \\ \vdots & \vdots & & \vdots & * & \cdots & & \cdots & * \\ 0 & 0 & \cdots & 1 & * & \cdots & & \cdots & * \\ 0 & 0 & \cdots & 0 & & & & & \\ \vdots & \vdots & & \vdots & & & \alpha_{r-d, d}(E) & & \\ 0 & 0 & \cdots & 0 & & & & & \end{array}\right),\quad E\in \mathcal{E}(h,0).\tag{8} \end{equation} This continues until \( r' = r - nd < d = d' + r' \). Then we apply (ii) and continue. Thus, for a given \( r \) and each \( d \), the set \( \mathcal{E}(r, d) \) is identified with the elliptic curve \( M \) and all steps of the infinite ladder, \( -\infty < d < \infty \), are almost identical. So far, we have introduced insufficient structure into the set \( \text{BunG} = G(F) \backslash \text{GL}(n, \mathbf{A}_F) / G(\mathcal{O}) \). There is topology but it is useless. The set in \( \text{BunG} \) is open if and only if its pre-image in \( \text{GL}(n, \mathbf{A}_F) / G(\mathcal{O}) \) is open. Nevertheless, there is a decomposition of \( \text{BunG} \) with respect to which its topology is largely unimportant. Namely, each sheaf is a direct sum of indecomposable sheaves of dimensions \( r_1, \ldots, r_k \), where \( r_1 + \ldots + r_k = r \). It turns out that for Hecke operators, the set \( \text{BunG} \) consists of individual sets \( \mathcal{D}(r_1, \ldots, r_k) \) according to the unordered set \( \{r_1, \ldots, r_k\} \), which defines the conjugacy class of Levi subgroups. Let \( \mathcal{E}(r) \) be the set of indecomposable sheaves of dimension \( r \), and \( \widetilde{\mathcal{E}}_k(r) \) be the symmetric \( k \)-fold product of \( \mathcal{E}(r) \) with itself. Then, \[ \mathcal{D}(r_1, \ldots, r_k) = \widetilde{\mathcal{E}}_{k_1}(s_1) \times \ldots \times \widetilde{\mathcal{E}}_{k_l}(s_l) \] The set \( \{r_1, \ldots , r_k\} \) consists of \( s_1 \) repetitions of \( k_1 \) and so on. Essentially, \( \mathcal{D}(r_1, \ldots , r_k) \) is the product set given by \[ \bigcup_{d=-\infty}^\infty \mathcal{E}(r, d),\tag{9} \] where \( r \) is given and \( \mathcal{E}(r, d) \) is the set of indecomposable sheaves of dimension \( r \) and degree \( d \). As a topological space, this is approximately \( \mathbb{Z} \times M = \mathbb{Z} \times U \times U \), where \( U = \{z \in \mathbb{C} \, | \, |z| = 1\} \). This is also a topological group, and its character group is \( U \times \mathbb{Z} \times \mathbb{Z} \). It can be assumed that it also approximately parameterizes the eigenfunctions of Hecke operators. Keeping this in mind, we move on to a hypothesis, but first, an encouraging remark. According to Theorem 7 in [1], if \( \chi \) is a character of the group \( \mathcal{E}(1, 0) \), then \( \chi \) determines a function with values in \( U \) over each \( E(r, d) \) in the set (9). If \( z \in U \), then the second function \( \eta_z : N \rightarrow z^d \). So it is possible that there is a simple or rather straightforward description of the complete set of eigenfunctions of the Hecke operators. We will provide it for \( r = 2 \). For an elliptic curve, \( g = 1 \), and the group \( \Gamma \) is generated by elements \( A \), \( B \), \( ABA^{-1}B^{-1} = 1 \neq 0 \in \mathbb{Z} \). According to the hypothesis, the eigenfunctions of Hecke operators for \( GL(n) \) correspond to representations of \( \Gamma_{\text{aut}} \) of dimension \( n \). The parabolic eigenfunctions correspond to irreducible representations. Let \( \rho \) be such a representation. Then \( \rho(1) = \zeta \in U \) and \( \rho(A)\rho(B)\rho(A)^{-1}\rho(B)^{-1} = \zeta I \). Since \( \det(\zeta I) = 1 \), \( \zeta \) is a root of unity. Let \( k \) be its order. Additionally, \( \rho(B) \) and \( \zeta \rho(B) \) are similar matrices. Let \( k \, | \, n \) be the order of \( \rho(B) \). The simplest example is $$ A=\lambda\left(\begin{array}{cccccc} 0 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ \vdots & \vdots & & & & \vdots \\ 1 & 0 & 0 & 0 & \cdots & 0 \end{array}\right), \quad B=\mu\left(\begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0 \\ 0 & \eta & 0 & \cdots & 0 \\ 0 & 0 & \eta^2 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & \eta^{k-1} \end{array}\right) $$ where \( \lambda, \mu \in \mathbb{C}^\times \), and \( \eta_k = 1 \). However, in this article, I only consider \( GL(2) \). In the next three sections, I consider Hecke operators for the group \( GL(2) \). The last section is the most important, the most remarkable, but the first and second sections contain the necessary preparation. 5. Correspondences of Hecke Dimensions