The endomorphism ring $\mathrm{End}(L)$ of a complex lattice $L$ is the subring of $\mathbb{C}$ consisting of all $\alpha \in \mathbb{C}$ such that $\alpha L \subseteq L$. Equivalently, an element $\alpha$ lies in $\mathrm{End}(L)$ iff multiplication by $\alpha$ preserves the additive subgroup underlying $L$.
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Characterization of membership in the endomorphism ring: $\alpha \in \mathrm{End}(L)$ iff $\alpha \cdot z \in L$ for every $z \in L$.
A complex lattice $L$ is a proper $\mathcal{O}$-ideal if its endomorphism ring equals $\mathcal{O}$ exactly (not just contains $\mathcal{O}$). This is the standard notion of proper ideal in the theory of orders in imaginary quadratic fields.
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Two proper $\mathcal{O}$-ideals $L_1$ and $L_2$ are equivalent if there exist nonzero $\alpha, \beta \in \mathcal{O}$ with $\alpha L_1 = \beta L_2$ as subsets of $\mathbb{C}$. This is the equivalence relation whose classes form the ideal class group of $\mathcal{O}$.
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Reflexivity of ideal equivalence: every lattice is equivalent to itself (witnessed by $\alpha = \beta = 1$).
Symmetry of ideal equivalence: if $\alpha L_1 = \beta L_2$, then $\beta L_2 = \alpha L_1$ (with the roles of $\alpha$ and $\beta$ swapped).
Transitivity of ideal equivalence: if $\alpha_1 L_1 = \beta_1 L_2$ and $\alpha_2 L_2 = \beta_2 L_3$, then $(\alpha_2\alpha_1) L_1 = (\beta_1\beta_2) L_3$.
The relation IsEquivalent 𝒪 is an equivalence relation on complex lattices.
Ideal equivalence implies homothety: if $\alpha L_1 = \beta L_2$, then $L_2 = (\beta^{-1}\alpha) L_1$, so the lattices are homothetic with factor $\beta^{-1}\alpha$.
The type of proper $\mathcal{O}$-ideals: pairs $(L, \mathrm{proof})$ where $L$ is a complex lattice, $L$ is a proper $\mathcal{O}$-ideal (its endomorphism ring equals $\mathcal{O}$), and $L$ is contained in $\mathcal{O}$ as a subset of $\mathbb{C}$.
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The setoid structure on proper $\mathcal{O}$-ideals induced by ideal equivalence; the quotient is the ideal class group of $\mathcal{O}$.
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The ideal class group $\mathrm{Cl}(\mathcal{O})$ of an order $\mathcal{O}$: the quotient of proper $\mathcal{O}$-ideals by the equivalence relation of being equal up to scaling by nonzero elements of $\mathcal{O}$. It is a finite abelian group whose order is the class number $h(\mathcal{O})$.
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The class of a proper $\mathcal{O}$-ideal in the ideal class group.
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Two proper ideals define the same class in $\mathrm{Cl}(\mathcal{O})$ iff they are equivalent as ideals.
The product of two proper $\mathcal{O}$-ideals: given $\mathfrak{a}$ with period pair $(\omega_1, \omega_2)$ and $\mathfrak{b}$ with period pair $(\omega_1', \omega_2')$, the product is the lattice with period pair $(\omega_1\omega_1', \omega_1\omega_2')$. The result is again a proper $\mathcal{O}$-ideal.
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The subring $\mathcal{O} \subseteq \mathbb{C}$ viewed as a complex lattice, when $\mathcal{O}$ is an order in an imaginary quadratic field. This is the lattice underlying the unit ideal $\mathcal{O}$ itself.
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The underlying set of subringAsComplexLattice 𝒪 coincides with the set of
elements of $\mathcal{O}$.
The endomorphism ring of $\mathcal{O}$ (viewed as a lattice) equals $\mathcal{O}$ itself, so $\mathcal{O}$ is a proper ideal of itself.
The unit element of the ideal class group: the proper $\mathcal{O}$-ideal $\mathcal{O}$ itself.
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If $L_{\mathrm{inv}}$ has period pair $(\bar\omega_1/N(\omega_1), \bar\omega_2/N(\omega_1))$ for a proper $\mathcal{O}$-ideal $\mathfrak{a}$, then $L_{\mathrm{inv}}$ is again a proper $\mathcal{O}$-ideal. This is the construction of the inverse ideal in the ideal class group.
The conjugate-scaled lattice $L_{\mathrm{inv}}$ is contained in $\mathcal{O}$;
together with conjugateScaledLattice_isProperIdeal this shows that it qualifies as
a proper $\mathcal{O}$-ideal.
The inverse of a proper $\mathcal{O}$-ideal $\mathfrak{a}$ with period pair $(\omega_1, \omega_2)$: the proper $\mathcal{O}$-ideal with period pair $(\bar\omega_1/N(\omega_1), \bar\omega_2/N(\omega_1))$, which represents the inverse class in $\mathrm{Cl}(\mathcal{O})$.
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For any proper $\mathcal{O}$-ideals $\mathfrak{a}, \mathfrak{b}$, the product $\mathfrak{a}\mathfrak{b}$ is equivalent to $\mathfrak{b}$ in the ideal class group; this is the homothety $\mathfrak{a}\mathfrak{b} = \omega_1(\mathfrak{a}) \cdot \mathfrak{b}$ underlying the definition.
Ideal multiplication respects the equivalence relation: if $\mathfrak{a}_1 \sim \mathfrak{a}_2$ and $\mathfrak{b}_1 \sim \mathfrak{b}_2$, then $\mathfrak{a}_1\mathfrak{b}_1 \sim \mathfrak{a}_2\mathfrak{b}_2$. This makes ideal multiplication well-defined on class group elements.
Associativity of ideal multiplication up to equivalence: $(\mathfrak{a}\mathfrak{b}) \mathfrak{c} \sim \mathfrak{a}(\mathfrak{b}\mathfrak{c})$.
Commutativity of ideal multiplication up to equivalence: $\mathfrak{a}\mathfrak{b} \sim \mathfrak{b}\mathfrak{a}$.
Left identity for ideal multiplication up to equivalence: $\mathcal{O} \cdot \mathfrak{a} \sim \mathfrak{a}$.
Inverse property for ideal multiplication up to equivalence: $\mathfrak{a}^{-1} \cdot \mathfrak{a} \sim \mathcal{O}$.
Ideal inversion respects the equivalence relation: equivalent ideals have equivalent inverses. This makes inversion well-defined on class group elements.
The commutative group structure on the ideal class group $\mathrm{Cl}(\mathcal{O})$. Multiplication, identity, and inversion are induced from the corresponding operations on proper $\mathcal{O}$-ideals via the quotient by ideal equivalence.
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Finiteness of the ideal class group: there exist finitely many proper ideal representatives that surject onto $\mathrm{Cl}(\mathcal{O})$. This is the classical finiteness theorem for orders in imaginary quadratic fields.
The ideal class group $\mathrm{Cl}(\mathcal{O})$ is a finite type.
The class number $h(\mathcal{O}) = |\mathrm{Cl}(\mathcal{O})|$ of an order $\mathcal{O}$.
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A subring $\mathcal{O} \subseteq \mathbb{C}$ is an order in an imaginary quadratic field if it contains some non-integer element $\tau$ that satisfies a monic integer quadratic polynomial $\tau^2 + b\tau + c = 0$ with negative discriminant $b^2 - 4c < 0$.
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The setoid structure on proper $\mathcal{O}$-ideals induced by homothety: two proper ideals are identified if one is a complex scalar multiple of the other.
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The set of homothety classes of proper $\mathcal{O}$-ideals. When $\mathcal{O}$ is an order in an imaginary quadratic field, this set is in canonical bijection with the ideal class group $\mathrm{Cl}(\mathcal{O})$.
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A proper $\mathcal{O}$-ideal is contained in $\mathcal{O}$: every element of the lattice underlying a proper ideal lies in $\mathcal{O}$.
The first basis vector $\omega_1$ of a proper $\mathcal{O}$-ideal lies in $\mathcal{O}$.
If $L_2 = c \cdot L_1$ (homothety) with $c \neq 0$ and both are proper $\mathcal{O}$-ideals, then $c \omega_1(L_1) \in \mathcal{O}$.
For proper ideals of an imaginary-quadratic order, homothety implies ideal
equivalence. Combined with the converse IsEquivalent.isHomothetic, this means
the two notions of equivalence coincide for proper ideals.
For proper ideals of an order in an imaginary quadratic field, ideal equivalence and homothety are equivalent notions.
For an order $\mathcal{O}$ in an imaginary quadratic field, the ideal class group is in canonical bijection with the set of homothety classes of proper $\mathcal{O}$-ideals.
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The endomorphism ring $\mathrm{End}(L)$ of any complex lattice $L$ is commutative, since it is a subring of $\mathbb{C}$.
Classification of endomorphism rings (Theorem 12.17): the endomorphism ring
of any complex lattice $L$ is either isomorphic to $\mathbb{Z}$ (constructor isZ),
or is an order in an imaginary quadratic field (constructor isOrderInImagQuadField),
in which case $L$ embeds into that order via $\alpha \mapsto \alpha\omega_1$.
- isZ {L : ComplexLattice} (h : ∀ α ∈ L.endomorphismRing, ∃ (n : ℤ), α = ↑n) : L.EndomorphismRingClassification
- isOrderInImagQuadField {L : ComplexLattice} (hIsOrder : IsImagQuadOrder L.endomorphismRing) (hLatticeEmbed : ∀ α ∈ L.endomorphismRing, ∃ (a' : ℤ) (b' : ℤ), ↑a' * L.ω₁ + ↑b' * L.ω₂ = α * L.ω₁) : L.EndomorphismRingClassification
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A complex lattice $L$ has complex multiplication iff its endomorphism ring contains some non-integer element (i.e. is strictly larger than $\mathbb{Z}$).
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Every integer $n \in \mathbb{Z}$ acts on $L$ by multiplication, so $\mathbb{Z} \subseteq \mathrm{End}(L)$ for any complex lattice $L$.
Every $\tau \in \mathrm{End}(L)$ satisfies the characteristic polynomial $\tau^2 - (a+d)\tau + (ad - bc) = 0$ where $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ is the matrix representation of $\tau$ acting on the lattice basis $(\omega_1, \omega_2)$.
The discriminant $(a + d)^2 - 4(ad - bc) = (\mathrm{tr}\,\tau)^2 - 4\det\tau$ of the characteristic polynomial of $\tau \in \mathrm{End}(L) \setminus \mathbb{Z}$ is negative. Hence $\tau$ generates an imaginary quadratic extension of $\mathbb{Q}$.
Order witness: if $\tau \in \mathrm{End}(L)$ is not an integer, then there exists an element of $\mathrm{End}(L)$ (namely $\tau$ itself) satisfying a monic integer quadratic polynomial with negative discriminant. This shows that $\mathrm{End}(L)$ is an order in an imaginary quadratic field.
Theorem 12.17 (classification of endomorphism rings): for every complex lattice $L$, either $\mathrm{End}(L) \cong \mathbb{Z}$ or $\mathrm{End}(L)$ is an order in an imaginary quadratic field.
Combined statement of Theorem 12.17: $\mathrm{End}(L)$ is commutative and is either $\mathbb{Z}$ or an order in an imaginary quadratic field.
Zero belongs to the lattice multiplier set $\{\alpha \in \mathbb{C} : \alpha L_1 \subseteq L_2\}$.
The lattice multiplier set is closed under addition: if $\alpha L_1, \beta L_1 \subseteq L_2$, then $(\alpha + \beta) L_1 \subseteq L_2$.
The lattice multiplier set is closed under negation: if $\alpha L_1 \subseteq L_2$, then $-\alpha \cdot L_1 \subseteq L_2$.
The lattice multiplier set $\{\alpha : \alpha L_1 \subseteq L_2\}$ packaged as an additive subgroup of $\mathbb{C}$.
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The induced map between complex tori is additive in the multiplier: the map induced by $\alpha + \beta$ equals the sum of the maps induced by $\alpha$ and $\beta$ separately.
Corollary 16.2, surjectivity part: every holomorphic map of complex tori sending $0$ to $0$ is induced by some multiplier $\alpha \in \mathbb{C}$ with $\alpha L_1 \subseteq L_2$.
Corollary 16.2, injectivity part: the multiplier $\alpha$ inducing a given holomorphic map of complex tori is uniquely determined.
Corollary 16.2 (combined): the set of holomorphic torus maps $L_1 \to L_2$ sending $0$ to $0$ is in additive bijection with the multiplier set $\{\alpha : \alpha L_1 \subseteq L_2\}$.
A function $f : \mathbb{C} \to \mathbb{C}$ is even iff $f(-z) = f(z)$ for all $z$.
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The composition $z \mapsto f(-z)$ of a meromorphic function with negation is again meromorphic.
If $f$ is $L$-periodic, then so is $z \mapsto f(-z)$.
The even part $f_+(z) = \tfrac{1}{2}(f(z) + f(-z))$ of an elliptic function $f$ is again an elliptic function.
The even part $f_+ = \tfrac{1}{2}(f + f \circ (-\mathrm{id}))$ is indeed an even function.
The odd-part quotient $\tfrac{f(z) - f(-z)}{2 \wp'(z)}$ is an elliptic function: since both the odd part of $f$ and $\wp'$ are odd elliptic functions, their ratio is an even elliptic function.
The odd-part quotient $\tfrac{f(z) - f(-z)}{2\wp'(z)}$ is an even function.
Even holomorphic elliptic functions are polynomials in $\wp$: if $f$ is an even, $L$-periodic, meromorphic function with no poles outside $L$, then $f(z) = P(\wp(z))$ for some polynomial $P \in \mathbb{C}[X]$.
Even elliptic functions are rational functions in $\wp$: if $f$ is an even elliptic function, then there exist polynomials $P, Q \in \mathbb{C}[X]$ with $Q \neq 0$ such that $Q(\wp(z)) f(z) = P(\wp(z))$, i.e.\ $f = P(\wp)/Q(\wp)$.
At a zero of $\wp'$, the odd part of an elliptic function $f$ vanishes: if $\wp'(z) = 0$ then $f(z) = f(-z)$. This is needed for the rational expression of arbitrary elliptic functions in $\wp$ and $\wp'$.
Assembly lemma for Lemma 16.3(i): given rational expressions for the even part of $f$ (via $P_e, Q_e$) and for the odd-part quotient $\tfrac{f - f(-\cdot)}{2\wp'}$ (via $P_o, Q_o$), one assembles a rational expression $f = (Q_o P_e + Q_e P_o \wp') / (Q_e Q_o)$ in $\wp$ and $\wp'$.
Theorem: the field of elliptic functions for $L$ equals $\mathbb{C}(\wp, \wp')$. Every elliptic function $f$ can be expressed as $f(z) = (P_1(\wp(z)) + P_2(\wp(z)) \wp'(z)) / Q(\wp(z))$ for polynomials $P_1, P_2, Q \in \mathbb{C}[X]$ with $Q \neq 0$.
$\wp_{L_2}(\alpha z)$ is a rational function in $\wp_{L_1}(z)$: there exist polynomials $u, v$ with $v \neq 0$ such that $v(\wp_{L_1}(z)) \wp_{L_2}(\alpha z) = u(\wp_{L_1}(z))$. Condition (2) of Theorem 16.4.
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The "isogeny" associated to $\alpha$ exists at the level of $\wp$-functions and their derivatives: both $\wp_{L_2}(\alpha z)$ and $\wp'_{L_2}(\alpha z)$ can be expressed rationally in $\wp_{L_1}$ and $\wp'_{L_1}$. Condition (3) of Theorem 16.4.
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Theorem 16.4: (1) $\Rightarrow$ (2). If $\alpha L_1 \subseteq L_2$, then $\wp_{L_2}(\alpha z)$ is a rational function in $\wp_{L_1}(z)$.
Theorem 16.4: (2) $\Rightarrow$ (3). If $\wp_{L_2}(\alpha z)$ is rational in $\wp_{L_1}$, then both $\wp_{L_2}(\alpha z)$ and $\wp'_{L_2}(\alpha z)$ have the required rational expressions, giving an isogeny.
Theorem 16.4: (3) $\Rightarrow$ (1). Existence of the rational isogeny formulae for $\alpha$ forces $\alpha L_1 \subseteq L_2$.
Theorem 16.4. The following are equivalent: (1) $\alpha L_1 \subseteq L_2$; (2) $\wp_{L_2}(\alpha z)$ is a rational function in $\wp_{L_1}(z)$; (3) the unique isogeny induced by $\alpha$ exists (i.e.\ both $\wp_{L_2}(\alpha z)$ and $\wp'_{L_2}(\alpha z)$ admit rational expressions in $\wp_{L_1}$ and $\wp'_{L_1}$).
Degree relation: if $v(\wp_{L_1}(z)) \wp_{L_2}(\alpha z) = u(\wp_{L_1}(z))$ expresses the isogeny induced by $\alpha$, then $\deg u = \deg v + 1$.
For an endomorphism $\alpha$ of a single lattice $L$, the norm $\alpha \overline{\alpha}$ equals the degree $\deg u$ of the rational $\wp$-expression of the induced isogeny.
The endomorphism multiplier set $\{\alpha : \alpha L \subseteq L\}$ is closed under multiplication, mirroring composition of endomorphisms.
Multiplicativity of the induced map: the endomorphism of $\mathbb{C}/L$ attached to $\alpha \beta$ equals the composition of the endomorphisms attached to $\alpha$ and $\beta$.
Corollary 16.5 (ring isomorphism). The set $\{\alpha : \alpha L \subseteq L\}$ with addition and multiplication of complex numbers is naturally isomorphic to the ring of holomorphic endomorphisms of $\mathbb{C}/L$: existence and uniqueness from Corollary 16.2 together with additivity and multiplicativity of the induced maps.
The Rosati involution on $\mathrm{End}(\mathbb{C}/L)$: for any endomorphism multiplier $\alpha$, the complex conjugate $\overline{\alpha}$ is also an endomorphism multiplier.
The dual endomorphism (Rosati involution) of $\alpha$: an element of the endomorphism multiplier set realising the complex conjugate $\overline{\alpha}$.
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The dual endomorphism is the complex conjugate of the original endomorphism.
The trace of an endomorphism: $\alpha + \overline{\alpha}$, an integer (more precisely, a real-integer) by Theorem 12.21.
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Corollary 16.5 (involution = conjugation). The Rosati involution on the endomorphism ring of $\mathbb{C}/L$ is exactly complex conjugation.