Shimura Varieties and Automorphic Forms

Shimura varieties are higher-dimensional generalizations of modular curves: certain quasi-projective complex algebraic varieties, defined over canonical number fields, that parametrize Hodge structures (often abelian varieties with extra structure) and carry commuting actions of

G (A_{f})

and Galois groups. Automorphic forms are functions on adelic groups transforming under arithmetic subgroups; their realization in the cohomology of Shimura varieties is the geometric heart of the Langlands program. Key result: the cohomology of a Shimura variety decomposes into pieces indexed by automorphic representations, and the Galois action on these pieces realizes the conjectural Langlands correspondence. Applies to: number theory, the Langlands program, modularity, motives. Originated with Shimura (1960s), formalized by Deligne (1970s).

A Shimura datum is a pair $(G, X)$ where $G$ is a reductive group over $Q$ and $X$ is a $G (R)$ -conjugacy class of homomorphisms $h : S \to G_{R}$ from the Deligne torus $S = Res_{C / R} G_{m}$ , satisfying axioms that make $X$ a finite union of Hermitian symmetric domains. The associated Shimura variety is the inverse limit (over compact open subgroups $K \subset G (A_{f})$ ) of the double-coset spaces $Sh_{K} (G, X) = G (Q) \ X \times G (A_{f}) / K$ . These matter because they geometrize automorphic forms — automorphic representations appear in the étale and coherent cohomology of Shimura varieties — and because they descend to canonical models over reflex fields, allowing Galois actions to interact with Hecke actions. A surprising fact: although defined analytically, these varieties are algebraic and arithmetic, and the eigenvalues of Frobenius at unramified primes (Eichler–Shimura, Langlands–Kottwitz) match Satake parameters of automorphic forms — the bridge by which modularity theorems (Fermat) are ultimately proved.

The Core Insight

A reductive group $G / Q$ together with a "complex structure datum" $X$ produces a tower of arithmetic locally symmetric varieties with three compatible structures: (1) they are moduli spaces of Hodge structures / abelian varieties; (2) they carry a Hecke action of $G (A_{f})$ whose eigenforms are automorphic representations; (3) they descend to canonical models over number fields so that Galois acts on cohomology. The miracle is that the Hecke eigenvalues (automorphic side) and Frobenius eigenvalues (Galois side) are matched — this is the geometric incarnation of reciprocity.

Formal Statement

The Deligne Torus and Shimura Data

Let $S = Res_{C / R} G_{m}$ be the Deligne torus, with $S (R) = C^{\times}$ . A real Hodge structure of weight $n$ on a vector space $V$ is equivalent to a representation $h : S \to GL (V_{R})$ .

A Shimura datum is a pair $(G, X)$ where $G$ is a connected reductive group over $Q$ and $X$ is a $G (R)$ -conjugacy class of homomorphisms

h : S ⟶ G_{R}

⟳

X

is recovered as

X = G (R) \cdot h_{0}

for any base point

h_{0}

; the stabilizer is a maximal compact (mod center), so

X ≅ G (R) / K_{\infty} Z

satisfying Deligne's axioms (SV1)–(SV3):

(SV1) The Hodge structure on $Lie (G)_{C}$ induced by $Ad \circ h$ is of type ${(- 1, 1), (0, 0), (1, - 1)}$ .
(SV2) $ad h (i)$ is a Cartan involution of $G_{R}^{ad}$ (ensuring $X$ is Hermitian symmetric).
(SV3) $G^{ad}$ has no $Q$ -factor on which the projection of $h$ is trivial.

Shimura Variety at Level $K$

For a compact open subgroup $K \subseteq G (A_{f})$ , the Shimura variety of level $K$ is the double coset space

Sh_{K} (G, X) = G (Q) \ (X \times G (A_{f}) / K)

⟳

G (Q)

acts diagonally on

X

(via the conjugation action realizing

X

) and on

G (A_{f}) / K

by left translation; the quotient is a finite disjoint union of arithmetic quotients

Γ_{i} \ X^{+}

When $K$ is small enough (neat), this is a smooth quasi-projective complex variety. The full Shimura variety is the limit

Sh (G, X) = K lim Sh_{K} (G, X),

⟳take the inverse limit over

K

ordered by inclusion; transition maps are finite étale;

G (A_{f})

acts on the right, recovering each finite level as a quotient.

carrying a continuous right $G (A_{f})$ -action with $Sh_{K} = Sh / K$ .

Reflex Field and Canonical Model

The conjugacy class $X$ determines a $G (C)$ -conjugacy class of cocharacters $μ_{h} : G_{m, C} \to G_{C}$ via $μ_{h} (z) = h_{C} (z, 1)$ . The reflex field $E = E (G, X)$ is the field of definition of this conjugacy class, a number field. The canonical model is a model of $Sh_{K} (G, X)$ over $E$ , characterized by the action on special points (CM points).

Existence of Canonical Models — Shimura, Deligne, Borovoi, Milne

Every Shimura datum $(G, X)$ admits a canonical model $Sh_{K} (G, X)_{E}$ over its reflex field $E$ , unique up to unique isomorphism, on which Hecke operators are defined over $E$ and special (CM) points map to algebraic points with prescribed Galois action given by reciprocity laws.

Automorphic Forms and the Cohomological Realization

An automorphic form for $G$ is a smooth, $K_{\infty}$ -finite, $z$ -finite, slowly increasing function $f : G (Q) \ G (A) \to C$ . The space $L_{disc}^{2} (G (Q) \ G (A))$ decomposes:

L_{disc}^{2} = π ⨁ m (π) π, π = π_{\infty} \otimes v ⨂^{'} π_{v}

⟳by the theory of automorphic representations (Gelbart–Jacquet), the discrete spectrum decomposes into irreducible admissible representations

π

G (A)

with finite multiplicities

m (π)

; the restricted tensor product is over places

v

The connection to geometry: for an irreducible representation $(ρ, V_{ρ})$ of $G$ giving a local system $V_{ρ}$ , Matsushima's formula relates de Rham cohomology to $(g, K)$ -cohomology:

H^{∙} (Sh_{K}, V_{ρ}) ≅ π ⨁ m (π) π_{f}^{K} \otimes H^{∙} (g, K_{\infty}; π_{\infty} \otimes V_{ρ})

⟳write cohomology as relative Lie algebra cohomology of the space of automorphic forms; the spectral decomposition then distributes over

π

, with

π_{f}^{K}

the

K

-fixed vectors carrying the Hecke action.

Why It Holds / Derivation

From Double Cosets to Locally Symmetric Spaces

Step 1. Fix a base point

h_{0} \in X

and let

X^{+}

be its connected component. The stabilizer of

h_{0}

G (R)^{+}

K_{\infty} Z (R)

where

K_{\infty}

is maximal compact; thus

X^{+} ≅ G (R)^{+} / K_{\infty} Z

, a Hermitian symmetric domain by (SV1)–(SV2). Step 2. Decompose

G (A_{f}) = ⨆_{i} G (Q)^{+} g_{i} K

into a finite set of double cosets (finiteness of class number, strong approximation when applicable). Then

Sh_{K} (G, X) = i ⨆ Γ_{i} \ X^{+}, Γ_{i} = G (Q)^{+} \cap g_{i} K g_{i}^{- 1}

⟳each double coset contributes one arithmetic quotient;

Γ_{i}

is an arithmetic subgroup commensurable with

G (Z)

, acting properly discontinuously on

X^{+}

Step 3. By the Baily–Borel theorem, each

Γ_{i} \ X^{+}

is a quasi-projective complex algebraic variety (its Satake/minimal compactification is projective and normal). Step 4. The Hecke action:

g \in G (A_{f})

induces

Sh_{K} \to Sh_{g^{- 1} K g}

, and composing with the projection gives a correspondence on

Sh_{K}

whenever

g

normalizes commensurable subgroups — these are the Hecke correspondences, algebraic over

E

. Step 5. Smoothness: when

K

is neat (no nontrivial torsion), the

Γ_{i}

act freely, so the quotient is smooth. This completes the construction of

Sh_{K}

as a smooth quasi-projective

E

-variety.

■

Eichler–Shimura: Frobenius Meets Hecke (Modular Curve Case)

Take $G = GL_{2}$ , $X = H^{\pm}$ , giving the modular curves $Y (N)$ .

Step 1. Weight-

k

cusp forms appear in coherent cohomology, and

H^{1}

of the local system

Sym^{k - 2}

realizes them: by the Eichler–Shimura isomorphism

H^{1} (Γ\ H, Sym^{k - 2}) ≅ S_{k} (Γ) \oplus \overline{S_{k} (Γ)} \oplus Eis

⟳harmonic representatives split into holomorphic and antiholomorphic parts plus an Eisenstein contribution; this is the Hodge decomposition of weight-

k

modular cohomology.

Step 2. Reduce the modular curve mod

p

(good reduction at

p ∤ N

). The Hecke operator

T_{p}

is realized as a correspondence; mod

p

it decomposes into the graph of Frobenius and its transpose. Step 3. The Eichler–Shimura congruence relation:

T_{p} \equiv Frob_{p} + p \cdot ⟨ p ⟩ Frob_{p}^{*} (mod p)

⟳in characteristic

p

the Hecke correspondence parametrizing

p

-isogenies splits as the Frobenius graph (étale part) plus its Verschiebung dual (connected part), giving the additive relation on cycles.

Step 4. Translate into Galois representations: for a newform

f

with Hecke eigenvalues

a_{p}

, the 2-dimensional

ℓ

-adic Galois representation

ρ_{f}

attached to

f

satisfies

tr ρ_{f} (Frob_{p}) = a_{p}, det ρ_{f} (Frob_{p}) = p^{k - 1} χ (p)

⟳the characteristic polynomial of Frobenius on

H_{\overset{e}{ˊ} t}^{1}

equals the Hecke polynomial

X^{2} - a_{p} X + p^{k - 1} χ (p)

by Step 3 plus comparison of étale and Betti cohomology.

This is the prototype matching automorphic data ( $a_{p}$ ) with Galois data (Frobenius eigenvalues). $■$

Goro Shimura | 1930–2019

Shimura's work in the 1950s–60s on complex multiplication, the arithmetic of modular forms, and what he called "the arithmetic theory of automorphic functions" laid the geometric foundations. He constructed canonical models for quotients of the upper half-plane and Hilbert/Siegel cases, established reciprocity laws at special points, and (with Taniyama) framed the conjecture linking elliptic curves to modular forms. Deligne later abstracted Shimura's constructions into the axiomatic framework of Shimura data $(G, X)$ in his 1971 Bourbaki seminar and 1979 Corvallis article, coining "Shimura variety." Shimura himself was famously ambivalent about the name attached to his work.

Building Intuition

Think of the modular curve as the "Rosetta Stone" example. The upper half-plane $H$ parametrizes complex elliptic curves $C / (Z + τ Z)$ ; quotienting by $SL_{2} (Z)$ identifies isomorphic curves, giving the moduli space $Y (1)$ . A Shimura variety generalizes this in two directions at once: replace the moduli problem (elliptic curves → abelian varieties with polarization, endomorphisms, level structure → general Hodge structures) and replace the group $SL_{2}$ by a general reductive $G$ .

The role of $h : S \to G_{R}$ is to encode "where the complex structure sits." For elliptic curves, $h$ specifies how $C^{\times}$ acts on the tangent space — the Hodge decomposition $H^{1} = H^{1, 0} \oplus H^{0, 1}$ . Varying $h$ over the conjugacy class $X$ literally varies the Hodge structure, sweeping out the period domain.

Why adeles? The finite adelic factor $G (A_{f})$ is bookkeeping for all level structures at once: each compact open $K$ corresponds to a level, and $⨂_{p} G (Q_{p})$ tracks $p$ -power level structure for every prime simultaneously. The Hecke operators are then symmetries permuting these level structures — the same operators whose eigenvalues are the Fourier coefficients of modular forms.

The Shimura variety as a bridge: analytic uniformization, Hecke action, and Galois descent over the reflex field

Worked Examples

The Modular Curve $Y_0(11)$ and the Elliptic Curve $11a$

Given:

G = GL_{2}

X = H^{\pm}

, level

K = K_{0} (11)

(matrices upper-triangular mod 11). The Shimura variety

Sh_{K} = Y_{0} (11)

is the modular curve of genus 1. Steps:

The space of weight-2 cusp forms $S_{2} (Γ_{0} (11))$ is 1-dimensional, spanned by the newform $f = q \prod_{n \geq 1} (1 - q^{n})^{2} (1 - q^{11 n})^{2} = q - 2 q^{2} - q^{3} + 2 q^{4} + q^{5} + \dots$
By Eichler–Shimura, $H_{\overset{e}{ˊ} t}^{1} (Y_{0} (11), Q_{ℓ})$ carries a 2-dimensional Galois representation $ρ_{f}$ with $tr ρ_{f} (Frob_{p}) = a_{p}$ .
Read off coefficients: $a_{2} = - 2$ , $a_{3} = - 1$ , $a_{5} = + 1$ , $a_{7} = - 2$ , $a_{13} = + 4$ .
Compare to the elliptic curve $E : y^{2} + y = x^{3} - x^{2} - 10 x - 20$ (Cremona $11 a$ ). Count points $# E (F_{p}) = p + 1 - a_{p}$ :

p = 2

# E (F_{2}) = 2 + 1 - (- 2) = 5

. -

p = 3

# E (F_{3}) = 3 + 1 - (- 1) = 5

. -

p = 5

# E (F_{5}) = 5 + 1 - 1 = 5

. -

p = 7

# E (F_{7}) = 7 + 1 - (- 2) = 10

. Boxed answer:

a_{p} (f) = p + 1 - # E (F_{p}) for all p \neq = 11; ρ_{f} ≅ ρ_{E, ℓ} (Shimura-Taniyama-Weil)

The automorphic eigenvalues equal the geometric Frobenius traces — the Shimura variety realizes the modularity of $E$ .

Siegel Modular Variety $\mathcal{A}_2$ and Dimension Count

Given:

G = GSp_{4}

X = H_{2}

(Siegel upper half-space of degree 2). The Shimura variety parametrizes principally polarized abelian surfaces. Steps:

Compute $dim_{C} H_{2}$ : it consists of symmetric complex $2 \times 2$ matrices $τ$ with $Im τ > 0$ . The number of independent entries is $(2 2 + 1) = 3$ .
So $dim_{C} A_{2} = 3$ , matching the dimension of the moduli space of abelian surfaces ( $dim A_{g} = g (g + 1) /2$ , here $g = 2$ ).
Reflex field: the cocharacter $μ$ for $GSp_{4}$ has reflex field $E = Q$ , so $A_{2}$ has a canonical model over $Q$ .
Hodge type: the relevant Hodge structure $H^{1}$ of an abelian surface has type ${(1, 0), (0, 1)}$ , weight 1, dimension 4 — consistent with (SV1).

Boxed answer:

dim_{C} A_{2} = 3, E (GSp_{4}, H_{2}) = Q

Siegel modular forms of degree 2 contribute to the coherent cohomology of $A_{2}$ and (via Langlands–Kottwitz) to 4-dimensional spin Galois representations.

Common Misconceptions

"Every Shimura variety is a moduli space of abelian varieties." False in general. Shimura varieties of Hodge type (those admitting an embedding into a Siegel datum) and PEL type (defined by polarization, endomorphisms, level structure) do parametrize abelian varieties with extra structure. But abelian-type and especially exceptional-type Shimura varieties (e.g. those for groups of type

E_{6}, E_{7}

) generally have no moduli interpretation in terms of motives one can directly write down — only an interpretation as parameter spaces for Hodge structures. The existence of canonical models for these required the indirect methods of Borovoi and Milne, not moduli theory. "The connected component $X^{+} /Γ$ is the full Shimura variety." The full

Sh_{K}

is a disjoint union of finitely many such quotients, indexed by

G (Q)^{+} \ G (A_{f}) / K

(a class group). Forgetting the adelic component loses the Hecke action and obscures the canonical model over

E

— and the reciprocity laws are statements about how Galois permutes the geometric components, encoded by the determinant/similitude character. Non-compactness and Eisenstein contamination. For

G = GL_{2}

the modular curve is non-compact (it has cusps). One might hope

H^{1} (Sh_{K}, Q)

is purely cuspidal, but it contains an Eisenstein part from the boundary. Concretely,

H^{1} (Y (1)) = 0

has no cusp forms, yet the compactly supported and ordinary cohomologies differ precisely by boundary (Eisenstein) classes. This forces the use of interior (parabolic) cohomology

H_{!}^{1} = im (H_{c}^{1} \to H^{1})

to isolate the cuspidal automorphic representations. Failing to do so leads to spurious "modular forms" that are merely Eisenstein series. The Borel–Serre and toroidal/minimal compactifications were developed precisely to control this boundary.

Connections

Shimura varieties sit at the crossroads of several theories. They are the geometric engine of the Langlands Program: the conjectural matching of automorphic representations with Galois representations is proved in many cases by decomposing étale cohomology (Langlands–Kottwitz method, Harris–Taylor, Scholze). Their construction depends on Hermitian Symmetric Spaces (the archimedean factor $X$ ) and Reductive Algebraic Groups (the group $G$ ). The arithmetic descent uses the theory of Complex Multiplication and Abelian Varieties (for PEL types), and the cohomological side connects to Hodge Theory via variations of Hodge structure over $X$ . The trace-formula comparison underlying point-counting connects to the Arthur–Selberg Trace Formula and Stable Trace Formula. The $p$ -adic geometry of these spaces (Rapoport–Zink, perfectoid Shimura varieties) ties into p-adic Hodge Theory and Perfectoid Spaces.

Real-World Applications

Modularity of elliptic curves / Fermat's Last Theorem (Wiles, Taylor–Wiles, 1995): The modular curve as a Shimura variety provides the Galois representations whose modularity, via $R = T$ theorems, proves semistable elliptic curves are modular — yielding FLT.

The Sato–Tate Conjecture (Clozel–Harris–Shepherd-Barron–Taylor, Taylor, 2008–11): Potential automorphy theorems use symmetric-power transfers realized in the cohomology of unitary Shimura varieties to prove equidistribution of Frobenius angles of elliptic curves.

Construction of Galois representations attached to automorphic forms (Harris–Taylor 2001, Shin, Scholze 2013): Unitary-group Shimura varieties produce the $ℓ$ -adic Galois representations attached to self-dual cuspidal representations of $GL_{n}$ , establishing the local Langlands correspondence for $GL_{n}$ over $p$ -adic fields.

The Birch–Swinnerton-Dyer Conjecture (partial results, Gross–Zagier, Kolyvagin): Heegner points — special (CM) points on modular curves (Shimura varieties for $GL_{2}$ ) — give the analytic-rank-1 cases of BSD via the Gross–Zagier formula relating heights to $L^{'} (E, 1)$ .

Bloch–Kato and Iwasawa Main Conjectures: Euler systems built from CM points and Beilinson–Flach elements live on towers of Shimura varieties and control Selmer groups.

Explicit reciprocity / class field theory of CM fields: Special values of automorphic forms at CM points on Shimura varieties generate abelian extensions (Shimura's reciprocity law), generalizing Kronecker's Jugendtraum.

Generalized modularity and potential automorphy in the proof of the Fontaine–Mazur conjecture cases: Cohomology of Shimura varieties supplies the automorphic lifts matching geometric Galois representations.

Historical Development

1957–1961 — Shimura & Taniyama: Develop the theory of complex multiplication of abelian varieties; the Taniyama–Shimura conjecture (every elliptic curve over $Q$ is modular) is formulated.
1959–1967 — Shimura: Series of papers on "the arithmetic of automorphic functions," constructing canonical models for quotients of $H$ and proving reciprocity laws at special points; introduces what become PEL-type spaces.
1966 — Eichler & Shimura: The Eichler–Shimura congruence relation $T_{p} \equiv F + p F^{*}$ established, linking Hecke operators to Frobenius on modular curves mod $p$ .
1971 — Deligne (Bourbaki 389): Reformulates Shimura's theory axiomatically via $(G, X)$ and $h : S \to G_{R}$ ; coins the framework of Shimura varieties; proves existence of canonical models in the Siegel (and Hodge-type) case.
1973–74 — Deligne: Proves the Weil conjectures, enabling control of weights in étale cohomology of Shimura varieties; "Travaux de Shimura."
1979 — Deligne (Corvallis), Langlands: "Variétés de Shimura"; Langlands's conjectures on the zeta functions of Shimura varieties and the point-counting program.
1980s — Kottwitz, Milne, Borovoi: Kottwitz computes points of PEL Shimura varieties over finite fields (Kottwitz conjecture); Borovoi and Milne establish canonical models for all Shimura varieties of abelian type.
2001 — Harris–Taylor: The Geometry and Cohomology of Some Simple Shimura Varieties proves local Langlands for $GL_{n}$ and constructs Galois representations.
2013–2015 — Scholze: Uses perfectoid Shimura varieties and the Hodge–Tate period map to construct Galois representations for torsion classes and over general number fields; revolutionizes the $p$ -adic theory.

Subtleties & Limits

Several deep problems remain. The Tate and Hodge conjectures for Shimura varieties are open in general — even understanding all the algebraic cycles on $A_{g}$ is unresolved. The functoriality of the cohomology under endoscopic transfer requires the full stabilization of the trace formula; while Arthur's work (conditional on twisted weighted fundamental lemmas, now mostly theorems via Ngô) handles classical groups, the general endoscopic classification for arbitrary $G$ is incomplete. For non-PEL, exceptional-type Shimura varieties (groups of type $E_{6}, E_{7}$ ), there is no known moduli interpretation, so constructing the attached motives and Galois representations directly is open. Integral models with good reduction at primes of bad/ramified reduction (the Langlands–Rapoport conjecture, Kisin–Pappas theory) are only partially understood, and the precise structure of the special fiber (local models, Rapoport–Zink spaces) for general parahoric level remains active research. Finally,