oxilean-std 0.1.2

//! Auto-generated module
//!
//! 🤖 Generated with [SplitRS](https://github.com/cool-japan/splitrs)
use super::functions::*;

/// A Chow group CH^n(X) of codimension-n algebraic cycles modulo rational equivalence.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct ChowGroup {
    /// The algebraic variety X (name).
    pub variety: String,
    /// The codimension n.
    pub codimension: usize,
    /// A list of generator descriptions.
    pub generators: Vec<String>,
}
#[allow(dead_code)]
impl ChowGroup {
    /// Create the Chow group CH^n(X).
    pub fn new(variety: impl Into<String>, codimension: usize) -> Self {
        Self {
            variety: variety.into(),
            codimension,
            generators: Vec::new(),
        }
    }
    /// CH^0(X) = Z (free on connected components).
    pub fn ch0(variety: impl Into<String>) -> Self {
        let mut g = Self::new(variety, 0);
        g.generators.push("[X]".to_string());
        g
    }
    /// CH^1(X) = Pic(X), the Picard group.
    pub fn pic(variety: impl Into<String>) -> Self {
        let mut g = Self::new(variety, 1);
        g.generators.push("Pic".to_string());
        g
    }
    /// Add a cycle class generator.
    pub fn add_generator(&mut self, gen: impl Into<String>) {
        self.generators.push(gen.into());
    }
    /// The intersection product CH^p × CH^q → CH^{p+q} (placeholder).
    pub fn intersect(&self, other: &Self) -> Self {
        Self {
            variety: self.variety.clone(),
            codimension: self.codimension + other.codimension,
            generators: vec![format!(
                "{} · {}",
                self.generators.first().map(|s| s.as_str()).unwrap_or("?"),
                other.generators.first().map(|s| s.as_str()).unwrap_or("?")
            )],
        }
    }
    /// Rank of the Chow group (number of generators in this model).
    pub fn rank(&self) -> usize {
        self.generators.len()
    }
}
/// A Berkovich space M(A) represented by a finite list of seminorm data.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct BerkovichSpace {
    /// The affinoid algebra A (name/description).
    pub algebra: String,
    /// The non-archimedean base field K.
    pub base_field: String,
    /// Dimension as a topological space (= Krull dim of A).
    pub dimension: usize,
    /// Whether this is a smooth Berkovich space.
    pub is_smooth: bool,
}
#[allow(dead_code)]
impl BerkovichSpace {
    /// Create a Berkovich space M(A) over K.
    pub fn new(algebra: impl Into<String>, base_field: impl Into<String>, dim: usize) -> Self {
        Self {
            algebra: algebra.into(),
            base_field: base_field.into(),
            dimension: dim,
            is_smooth: false,
        }
    }
    /// The Berkovich unit disc over K: M(K{T}).
    pub fn unit_disc(base_field: impl Into<String>) -> Self {
        Self {
            algebra: "K{T}".to_string(),
            base_field: base_field.into(),
            dimension: 1,
            is_smooth: true,
        }
    }
    /// The Berkovich affine n-space over K: M(K{T_1,...,T_n}).
    pub fn affine_n_space(base_field: impl Into<String>, n: usize) -> Self {
        let gens = (1..=n)
            .map(|i| format!("T_{}", i))
            .collect::<Vec<_>>()
            .join(", ");
        Self {
            algebra: format!("K{{{}}}", gens),
            base_field: base_field.into(),
            dimension: n,
            is_smooth: true,
        }
    }
    /// The Gauss point (the canonical point in M(K{T})).
    pub fn gauss_point_description(&self) -> String {
        format!(
            "Gauss point of M({}) over {}: seminorm |·|_1",
            self.algebra, self.base_field
        )
    }
    /// Mark as smooth.
    pub fn smooth(mut self) -> Self {
        self.is_smooth = true;
        self
    }
    /// The skeleton (dual graph / tropical variety) description.
    pub fn skeleton_description(&self) -> String {
        format!("Skeleton of M({}) (dim ≤ {})", self.algebra, self.dimension)
    }
}
/// Absolute Weil height H(P) on projective space.
#[derive(Debug, Clone)]
pub struct AbsoluteHeight {
    /// Description of the point.
    pub point_description: String,
    /// The computed height value.
    pub value: f64,
}
impl AbsoluteHeight {
    /// Compute the absolute height of a rational number p/q in lowest terms.
    ///
    /// H(p/q) = max(|p|, |q|) for \[p:q\] ∈ P^1(ℚ).
    pub fn of_rational(p: i64, q: u64) -> Self {
        let value = (p.unsigned_abs()).max(q) as f64;
        Self {
            point_description: format!("{}/{}", p, q),
            value,
        }
    }
    /// Height of an algebraic number from its minimal polynomial coefficients.
    pub fn of_minimal_poly(coeffs: &[i64]) -> Self {
        let max_coeff = coeffs.iter().map(|c| c.unsigned_abs()).max().unwrap_or(0);
        Self {
            point_description: format!("root of poly with coeffs {:?}", coeffs),
            value: max_coeff as f64,
        }
    }
    /// Height value.
    pub fn value(&self) -> f64 {
        self.value
    }
}
/// The Langlands correspondence ρ_π ↔ π.
#[derive(Debug, Clone)]
pub struct LanglandsCorrespondence {
    /// The Galois representation.
    pub galois_rep: GaloisRepresentation,
    /// The automorphic representation.
    pub automorphic_rep: AutomorphicRepresentation,
    /// Which case of Langlands is being invoked.
    pub correspondence_type: LanglandsType,
}
impl LanglandsCorrespondence {
    /// Create a Langlands correspondence.
    pub fn new(
        gal: GaloisRepresentation,
        aut: AutomorphicRepresentation,
        kind: LanglandsType,
    ) -> Self {
        Self {
            galois_rep: gal,
            automorphic_rep: aut,
            correspondence_type: kind,
        }
    }
}
/// The Northcott property: finitely many points of bounded height.
#[derive(Debug, Clone)]
pub struct NorthcottProperty {
    /// The space or set satisfying Northcott.
    pub space: String,
    /// Whether this is an absolute degree-and-height bound.
    pub absolute: bool,
}
impl NorthcottProperty {
    /// Projective space P^n satisfies Northcott over any number field.
    pub fn projective_space(n: usize) -> Self {
        Self {
            space: format!("P^{}", n),
            absolute: true,
        }
    }
    /// Check finitely-many-points-of-bounded-height property for a given bound.
    ///
    /// Returns an approximate count estimate for P^1(ℚ) with H ≤ B.
    pub fn count_estimate_p1_rational(bound: f64) -> usize {
        if bound < 1.0 {
            return 0;
        }
        ((12.0 / std::f64::consts::PI.powi(2)) * bound * bound).round() as usize
    }
}
/// An abelian variety over a field k of dimension g.
#[derive(Debug, Clone, PartialEq, Eq)]
pub struct AbelianVariety {
    /// The base field (name).
    pub field: String,
    /// The dimension g (genus for abelian surfaces g=2, etc.).
    pub dimension: usize,
    /// Whether this is a simple abelian variety.
    pub is_simple: bool,
    /// Label/name.
    pub name: String,
}
impl AbelianVariety {
    /// Create an abelian variety over k of dimension g.
    pub fn new(field: impl Into<String>, dimension: usize) -> Self {
        Self {
            field: field.into(),
            dimension,
            is_simple: false,
            name: String::new(),
        }
    }
    /// Named abelian variety.
    pub fn named(field: impl Into<String>, dimension: usize, name: impl Into<String>) -> Self {
        Self {
            field: field.into(),
            dimension,
            is_simple: false,
            name: name.into(),
        }
    }
    /// Dimension of this abelian variety.
    pub fn dimension(&self) -> usize {
        self.dimension
    }
    /// Rank of the Tate module T_p(A) = 2g.
    pub fn tate_module_rank(&self) -> usize {
        2 * self.dimension
    }
    /// Trace of Frobenius on T_p(A) (placeholder).
    pub fn trace_of_frobenius(&self, _prime: u64) -> i64 {
        0
    }
    /// Endomorphism algebra description.
    pub fn endomorphism_ring(&self) -> String {
        if self.is_simple {
            "Division algebra".to_string()
        } else {
            "Matrix algebra".to_string()
        }
    }
    /// Mark as simple.
    pub fn simple(mut self) -> Self {
        self.is_simple = true;
        self
    }
}
/// An automorphic representation π = ⊗_v π_v.
#[derive(Debug, Clone)]
pub struct AutomorphicRepresentation {
    /// The reductive group G.
    pub group: String,
    /// The number field F.
    pub field: String,
    /// The infinitesimal character (weight).
    pub weight: Vec<i32>,
    /// Whether this is cuspidal.
    pub is_cuspidal: bool,
}
impl AutomorphicRepresentation {
    /// Create an automorphic representation of G over F.
    pub fn new(group: impl Into<String>, field: impl Into<String>) -> Self {
        Self {
            group: group.into(),
            field: field.into(),
            weight: Vec::new(),
            is_cuspidal: false,
        }
    }
    /// A classical holomorphic modular form of weight k (automorphic for GL_2).
    pub fn modular_form_gl2(weight: i32, field: impl Into<String>) -> Self {
        Self {
            group: "GL_2".to_string(),
            field: field.into(),
            weight: vec![weight],
            is_cuspidal: true,
        }
    }
    /// Tensor product of local components at a finite set of places.
    pub fn local_components(&self) -> String {
        format!("⊗_v π_v of {} over {}", self.group, self.field)
    }
}
/// Type of Langlands correspondence.
#[derive(Debug, Clone, PartialEq, Eq)]
pub enum LanglandsType {
    /// Local Langlands for GL_n (proved by Harris-Taylor and Henniart).
    LocalGLn,
    /// Global Langlands for GL_2 over ℚ (class field theory + modular forms).
    GlobalGL2,
    /// Functoriality transfer.
    Functoriality,
    /// Geometric Langlands.
    Geometric,
}
/// Logarithmic (Weil) height h(P) = log H(P).
#[derive(Debug, Clone)]
pub struct LogarithmicHeight {
    /// The absolute height.
    pub absolute: AbsoluteHeight,
}
impl LogarithmicHeight {
    /// Create from an absolute height.
    pub fn from_absolute(h: AbsoluteHeight) -> Self {
        Self { absolute: h }
    }
    /// Compute h(p/q) = log max(|p|, |q|) for \[p:q\] ∈ P^1(ℚ).
    pub fn of_rational(p: i64, q: u64) -> Self {
        Self::from_absolute(AbsoluteHeight::of_rational(p, q))
    }
    /// The logarithmic height value.
    pub fn value(&self) -> f64 {
        self.absolute.value.ln()
    }
}
/// A condensed abelian group: a sheaf of abelian groups on the category of profinite sets.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct CondensedAbelianGroup {
    /// Descriptive label (e.g., "R", "Z", "Q/Z").
    pub label: String,
    /// Whether this group is solid (satisfies the solid tensor product condition).
    pub is_solid: bool,
    /// Whether this is a discrete abelian group (embedded into condensed ab. groups).
    pub is_discrete: bool,
}
#[allow(dead_code)]
impl CondensedAbelianGroup {
    /// Wrap a discrete abelian group as a condensed group.
    pub fn discrete(label: impl Into<String>) -> Self {
        Self {
            label: label.into(),
            is_solid: false,
            is_discrete: true,
        }
    }
    /// The solid abelian group Z\[S\] for a profinite set S.
    pub fn solid_free(profinite_set: impl Into<String>) -> Self {
        Self {
            label: format!("Z[{}]^solid", profinite_set.into()),
            is_solid: true,
            is_discrete: false,
        }
    }
    /// Mark as solid.
    pub fn solidify(mut self) -> Self {
        self.is_solid = true;
        self
    }
    /// The solid tensor product A ⊗_solid B (placeholder returning description).
    pub fn solid_tensor_product(&self, other: &Self) -> String {
        format!("{} ⊗_solid {}", self.label, other.label)
    }
    /// Check if the group satisfies the liquid vector space condition for exponent p.
    pub fn is_p_liquid(&self, p: f64) -> bool {
        self.is_solid && p > 0.0 && p <= 1.0
    }
}
/// A Néron model data record for an abelian variety A over the fraction field of a DVR.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct NeronModel {
    /// The abelian variety A (name).
    pub variety: String,
    /// The discrete valuation ring R (e.g., "Z_p", "O_K").
    pub dvr: String,
    /// The fraction field K = Frac(R).
    pub fraction_field: String,
    /// Reduction type of the Néron model special fiber.
    pub reduction_type: NeronReductionType,
    /// The component group Φ = A_s^0\A_s of the special fiber.
    pub component_group_order: Option<u64>,
}
#[allow(dead_code)]
impl NeronModel {
    /// Create a Néron model record.
    pub fn new(
        variety: impl Into<String>,
        dvr: impl Into<String>,
        fraction_field: impl Into<String>,
        reduction_type: NeronReductionType,
    ) -> Self {
        Self {
            variety: variety.into(),
            dvr: dvr.into(),
            fraction_field: fraction_field.into(),
            reduction_type,
            component_group_order: None,
        }
    }
    /// Néron model with good reduction.
    pub fn good_reduction(
        variety: impl Into<String>,
        dvr: impl Into<String>,
        frac: impl Into<String>,
    ) -> Self {
        Self::new(variety, dvr, frac, NeronReductionType::Good)
    }
    /// Néron model with semi-stable reduction.
    pub fn semi_stable(
        variety: impl Into<String>,
        dvr: impl Into<String>,
        frac: impl Into<String>,
    ) -> Self {
        Self::new(variety, dvr, frac, NeronReductionType::SemiStable)
    }
    /// Set the component group order |Φ|.
    pub fn with_component_group(mut self, order: u64) -> Self {
        self.component_group_order = Some(order);
        self
    }
    /// Whether the Néron model has good reduction.
    pub fn has_good_reduction(&self) -> bool {
        self.reduction_type == NeronReductionType::Good
    }
    /// Whether the Néron model has semi-stable reduction.
    pub fn is_semi_stable(&self) -> bool {
        matches!(
            self.reduction_type,
            NeronReductionType::Good
                | NeronReductionType::SemiStable
                | NeronReductionType::PurelyToric
        )
    }
    /// Tamagawa number c_v = |Φ(k_v)| (number of connected components over residue field).
    pub fn tamagawa_number(&self) -> u64 {
        self.component_group_order.unwrap_or(1)
    }
}
/// Reduction type of the special fiber of a Néron model.
#[allow(dead_code)]
#[derive(Debug, Clone, PartialEq, Eq)]
pub enum NeronReductionType {
    /// Good reduction: A_s is an abelian variety.
    Good,
    /// Semi-stable reduction: A_s^0 is a semi-abelian variety.
    SemiStable,
    /// Purely toric reduction: A_s^0 is a torus.
    PurelyToric,
    /// Additive reduction.
    Additive,
}
/// An elliptic curve E: y² = x³ + ax + b over a field k.
#[derive(Debug, Clone)]
pub struct EllipticCurve {
    /// The base field.
    pub field: String,
    /// Coefficient a.
    pub a: i64,
    /// Coefficient b.
    pub b: i64,
}
impl EllipticCurve {
    /// Create an elliptic curve y² = x³ + ax + b.
    pub fn new(field: impl Into<String>, a: i64, b: i64) -> Self {
        Self {
            field: field.into(),
            a,
            b,
        }
    }
    /// Discriminant Δ = -16(4a³ + 27b²). Non-zero iff the curve is non-singular.
    pub fn discriminant(&self) -> i64 {
        -16 * (4 * self.a.pow(3) + 27 * self.b.pow(2))
    }
    /// Whether the curve is non-singular (Δ ≠ 0).
    pub fn is_non_singular(&self) -> bool {
        self.discriminant() != 0
    }
    /// j-invariant j = -1728 * (4a)³ / Δ (simplified for short Weierstrass).
    pub fn j_invariant(&self) -> Option<f64> {
        let delta = self.discriminant();
        if delta == 0 {
            return None;
        }
        let numerator = -1728.0 * (4.0 * (self.a as f64)).powi(3);
        Some(numerator / (delta as f64))
    }
    /// Check if this is an isomorphism class representative (j-invariant determines it over alg. closed).
    pub fn j_class(&self) -> String {
        match self.j_invariant() {
            Some(j) => format!("j = {:.4}", j),
            None => "Singular (not an elliptic curve)".to_string(),
        }
    }
}
/// The p-adic Tate module T_p(A) = lim A[p^n].
#[derive(Debug, Clone, PartialEq, Eq)]
pub struct TateModule {
    /// The abelian variety (name).
    pub variety: String,
    /// The prime p.
    pub prime: u64,
    /// Rank over ℤ_p (= 2g for an abelian variety of dimension g).
    pub rank: usize,
}
impl TateModule {
    /// Create T_p(A) for an abelian variety of dimension g.
    pub fn new(variety: impl Into<String>, prime: u64, dimension: usize) -> Self {
        Self {
            variety: variety.into(),
            prime,
            rank: 2 * dimension,
        }
    }
}
/// A Shimura datum (G, X).
#[derive(Debug, Clone)]
pub struct ShimuraDatum {
    /// The reductive group G (name, e.g., "GSp_{2g}", "GL_2", "U(p,q)").
    pub group: String,
    /// The hermitian symmetric domain X (e.g., "Siegel upper half-space").
    pub domain: String,
    /// The reflex field E(G, X).
    pub reflex_field: String,
}
impl ShimuraDatum {
    /// Create a Shimura datum.
    pub fn new(
        group: impl Into<String>,
        domain: impl Into<String>,
        reflex_field: impl Into<String>,
    ) -> Self {
        Self {
            group: group.into(),
            domain: domain.into(),
            reflex_field: reflex_field.into(),
        }
    }
    /// The Siegel Shimura datum (GL_2, H): modular curves.
    pub fn gl2_upper_half_plane() -> Self {
        Self::new("GL_2", "H (upper half-plane)", "Q")
    }
    /// The Siegel Shimura datum (GSp_{2g}, H_g): Siegel modular varieties.
    pub fn siegel(g: usize) -> Self {
        Self::new(
            format!("GSp_{{2{}}}", g),
            format!("H_{} (Siegel half-space)", g),
            "Q",
        )
    }
}
/// Canonical model of a Shimura variety over its reflex field.
#[derive(Debug, Clone)]
pub struct CanonicalModel {
    /// The Shimura variety.
    pub shimura_variety: ShimuraVariety,
    /// The reflex field.
    pub reflex_field: String,
}
impl CanonicalModel {
    /// Create a canonical model.
    pub fn new(variety: ShimuraVariety, reflex_field: impl Into<String>) -> Self {
        Self {
            shimura_variety: variety,
            reflex_field: reflex_field.into(),
        }
    }
}
/// A nearly ordinary p-adic Galois representation (Borel reduction at p).
#[derive(Debug, Clone)]
pub struct NearlyOrdinaryRepresentation {
    /// The base representation.
    pub rep: GaloisRepresentation,
    /// The prime p at which the representation is nearly ordinary.
    pub prime: u64,
}
impl NearlyOrdinaryRepresentation {
    /// Create a nearly ordinary representation.
    pub fn new(rep: GaloisRepresentation, prime: u64) -> Self {
        Self { rep, prime }
    }
}
/// The canonical Néron-Tate height ĥ: E(K̄) → ℝ.
#[derive(Debug, Clone)]
pub struct HeightFunction {
    /// The elliptic curve.
    pub curve: String,
    /// The number field K over which points are defined.
    pub field: String,
}
impl HeightFunction {
    /// Create the Néron-Tate height for E over K.
    pub fn neron_tate(curve: impl Into<String>, field: impl Into<String>) -> Self {
        Self {
            curve: curve.into(),
            field: field.into(),
        }
    }
    /// Compute height of a rational point (a/d, b/d²) approximation.
    ///
    /// Uses h(x/y) ≈ (1/2) log max(|x|, |y|) as a rough naive height.
    pub fn naive_height(x_num: i64, x_den: u64) -> f64 {
        if x_den == 0 {
            return 0.0;
        }
        0.5 * ((x_num.unsigned_abs()).max(x_den) as f64).ln()
    }
}
/// A perfectoid field K with its residue characteristic p.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct PerfectoidField {
    /// Description of the field (e.g., "C_p", "Q_p^{\infty}").
    pub name: String,
    /// Residue characteristic p.
    pub residue_char: u64,
    /// Whether the field is algebraically closed (e.g., C_p).
    pub is_alg_closed: bool,
    /// Tilt field K^♭ (described by name).
    pub tilt_name: String,
}
#[allow(dead_code)]
impl PerfectoidField {
    /// Create a perfectoid field.
    pub fn new(name: impl Into<String>, residue_char: u64, tilt_name: impl Into<String>) -> Self {
        Self {
            name: name.into(),
            residue_char,
            is_alg_closed: false,
            tilt_name: tilt_name.into(),
        }
    }
    /// The completed algebraic closure C_p of Q_p.
    pub fn c_p(p: u64) -> Self {
        Self {
            name: format!("C_{}", p),
            residue_char: p,
            is_alg_closed: true,
            tilt_name: format!("C_{}^flat", p),
        }
    }
    /// The infinite cyclotomic extension Q_p(ζ_{p^∞}).
    pub fn q_p_cyclotomic(p: u64) -> Self {
        Self {
            name: format!("Q_{}(zeta_{{p^inf}})", p),
            residue_char: p,
            is_alg_closed: false,
            tilt_name: format!("F_{}((t^{{1/p^inf}}))", p),
        }
    }
    /// Residue characteristic of this perfectoid field.
    pub fn residue_characteristic(&self) -> u64 {
        self.residue_char
    }
    /// The tilt K^♭ has characteristic p.
    pub fn tilt_characteristic(&self) -> u64 {
        self.residue_char
    }
    /// Description of the Hodge-Tate decomposition for a de Rham representation.
    pub fn hodge_tate_description(&self, dim: usize) -> String {
        format!(
            "H^{{}} decomposes over {} as direct sum of dim {} C_p-spaces with Hodge-Tate weights",
            self.name, dim
        )
    }
}
/// An isogeny φ: E → E' (group homomorphism with finite kernel).
#[derive(Debug, Clone)]
pub struct Isogeny {
    /// Source elliptic curve.
    pub source: String,
    /// Target elliptic curve.
    pub target: String,
    /// Degree of the isogeny (= |ker φ|).
    pub degree: u64,
}
impl Isogeny {
    /// Create an isogeny of the given degree.
    pub fn new(source: impl Into<String>, target: impl Into<String>, degree: u64) -> Self {
        Self {
            source: source.into(),
            target: target.into(),
            degree,
        }
    }
    /// The multiplication-by-n map \[n\]: E → E (isogeny of degree n²).
    pub fn multiplication_by_n(curve: impl Into<String>, n: u64) -> Self {
        let c = curve.into();
        Self {
            source: c.clone(),
            target: c,
            degree: n * n,
        }
    }
    /// Whether this is an endomorphism (source = target).
    pub fn is_endomorphism(&self) -> bool {
        self.source == self.target
    }
}
/// The dual abelian variety A^ = Pic^0(A).
#[derive(Debug, Clone)]
pub struct DualAbelianVariety {
    /// The original variety.
    pub variety: AbelianVariety,
}
impl DualAbelianVariety {
    /// Compute the dual of an abelian variety.
    pub fn of(variety: AbelianVariety) -> Self {
        Self { variety }
    }
    /// Dimension of the dual variety (same as original).
    pub fn dimension(&self) -> usize {
        self.variety.dimension
    }
}
/// A polarized abelian variety: (A, λ) where λ: A → A^ is an isogeny from A to its dual.
#[derive(Debug, Clone)]
pub struct PolarizedAbelianVariety {
    /// The underlying abelian variety.
    pub variety: AbelianVariety,
    /// Degree of the polarization λ.
    pub polarization_degree: u64,
    /// Whether the polarization is principal (degree 1).
    pub is_principal: bool,
}
impl PolarizedAbelianVariety {
    /// Create a principally polarized abelian variety.
    pub fn principally_polarized(variety: AbelianVariety) -> Self {
        Self {
            variety,
            polarization_degree: 1,
            is_principal: true,
        }
    }
    /// Create a polarized abelian variety with given degree.
    pub fn polarized(variety: AbelianVariety, degree: u64) -> Self {
        Self {
            variety,
            polarization_degree: degree,
            is_principal: degree == 1,
        }
    }
}
/// Data for the Birch and Swinnerton-Dyer conjecture.
#[derive(Debug, Clone)]
pub struct BirchSwinnertonDyerData {
    /// The elliptic curve.
    pub curve: String,
    /// Rank of E(ℚ) (Mordell-Weil rank).
    pub rank: usize,
    /// Regulator (determinant of the Néron-Tate height pairing matrix).
    pub regulator: f64,
    /// Order of the Shafarevich-Tate group (conjectured finite).
    pub sha_order: Option<u64>,
    /// Whether BSD rank conjecture is verified for this curve.
    pub bsd_rank_verified: bool,
}
impl BirchSwinnertonDyerData {
    /// Create BSD data for an elliptic curve.
    pub fn new(curve: impl Into<String>, rank: usize, regulator: f64) -> Self {
        Self {
            curve: curve.into(),
            rank,
            regulator,
            sha_order: None,
            bsd_rank_verified: false,
        }
    }
    /// BSD formula prediction for the leading coefficient of L(E, s) at s=1.
    ///
    /// L^*(E, 1) = (|Sha| · Ω_E · ∏c_p · Reg) / |E(ℚ)_tors|²
    /// (simplified placeholder).
    pub fn leading_coefficient_prediction(&self) -> f64 {
        let sha = self.sha_order.unwrap_or(1) as f64;
        sha * self.regulator
    }
}
/// A Shimura variety Sh_K(G, X) as a moduli space.
#[derive(Debug, Clone)]
pub struct ShimuraVariety {
    /// The Shimura datum.
    pub datum: ShimuraDatum,
    /// The level K (compact open subgroup, described as a string).
    pub level: String,
    /// Complex dimension.
    pub complex_dimension: usize,
}
impl ShimuraVariety {
    /// Create a Shimura variety with the given datum and level.
    pub fn new(datum: ShimuraDatum, level: impl Into<String>, dim: usize) -> Self {
        Self {
            datum,
            level: level.into(),
            complex_dimension: dim,
        }
    }
    /// The modular curve Y(N) = Sh_{Γ(N)}(GL_2, H).
    pub fn modular_curve_y(n: usize) -> Self {
        Self::new(ShimuraDatum::gl2_upper_half_plane(), format!("Γ({})", n), 1)
    }
}
/// A continuous Galois representation ρ: G_K → GL_n(R).
#[derive(Debug, Clone)]
pub struct GaloisRepresentation {
    /// The Galois group G_K (described by the field K).
    pub galois_group: String,
    /// The dimension n.
    pub dimension: usize,
    /// The coefficient ring R (e.g., "Z_ell", "Q_ell", "C").
    pub coefficient_ring: String,
    /// Whether the representation is irreducible.
    pub is_irreducible: bool,
    /// Whether the representation is geometric (de Rham, Hodge-Tate).
    pub is_geometric: bool,
}
impl GaloisRepresentation {
    /// Create a Galois representation.
    pub fn new(gal_group: impl Into<String>, dim: usize, ring: impl Into<String>) -> Self {
        Self {
            galois_group: gal_group.into(),
            dimension: dim,
            coefficient_ring: ring.into(),
            is_irreducible: false,
            is_geometric: false,
        }
    }
    /// The ℓ-adic cyclotomic character χ_ℓ: G_ℚ → ℤ_ℓ^×.
    pub fn cyclotomic(ell: u64) -> Self {
        let mut r = Self::new("G_Q", 1, format!("Z_{}", ell));
        r.is_irreducible = true;
        r.is_geometric = true;
        r
    }
    /// The ℓ-adic Tate module representation of an elliptic curve E.
    pub fn from_elliptic_curve(curve: &str, ell: u64) -> Self {
        let mut r = Self::new("G_Q", 2, format!("Z_{}", ell));
        r.galois_group = format!("G_Q (from {})", curve);
        r.is_geometric = true;
        r
    }
    /// Mark as irreducible.
    pub fn irreducible(mut self) -> Self {
        self.is_irreducible = true;
        self
    }
}
/// Status record for the André-Oort conjecture (Tolimani's theorem).
#[derive(Debug, Clone)]
pub struct TolimaniConjecture {
    /// Proved under GRH (Klingler-Yafaev, 2014) and unconditionally (Pila-Tsimerman).
    pub proved: bool,
    /// Description of the proof method.
    pub proof_method: String,
}
impl TolimaniConjecture {
    /// The proved André-Oort conjecture.
    pub fn proved() -> Self {
        Self {
            proved: true,
            proof_method: "o-minimality (Pila-Wilkie) + height bounds".to_string(),
        }
    }
}
/// The n-torsion subgroup E\[n\] ⊂ E.
#[derive(Debug, Clone, PartialEq, Eq)]
pub struct TorsionPoint {
    /// The elliptic curve.
    pub curve: String,
    /// The torsion order n.
    pub n: u64,
    /// Structure: over an algebraically closed field of char 0, E\[n\] ≅ (ℤ/nℤ)².
    pub is_full_n_torsion: bool,
}
impl TorsionPoint {
    /// Full n-torsion over an algebraically closed field.
    pub fn full(curve: impl Into<String>, n: u64) -> Self {
        Self {
            curve: curve.into(),
            n,
            is_full_n_torsion: true,
        }
    }
    /// Size of the n-torsion group: |E\[n\]| = n² (over alg. closed field, char 0).
    pub fn size(&self) -> u64 {
        if self.is_full_n_torsion {
            self.n * self.n
        } else {
            self.n
        }
    }
}
/// Faltings's theorem (Mordell conjecture): curves of genus ≥ 2 have finitely many rational points.
#[derive(Debug, Clone)]
pub struct FaltingsThm {
    /// The algebraic curve (name).
    pub curve: String,
    /// The genus.
    pub genus: usize,
    /// The number field.
    pub field: String,
    /// Known rational points.
    pub known_rational_points: Vec<String>,
}
impl FaltingsThm {
    /// Create a Faltings theorem application for a curve of genus g.
    pub fn for_curve(curve: impl Into<String>, genus: usize, field: impl Into<String>) -> Self {
        Self {
            curve: curve.into(),
            genus,
            field: field.into(),
            known_rational_points: Vec::new(),
        }
    }
    /// Whether finiteness of rational points is guaranteed by Faltings.
    pub fn finiteness_guaranteed(&self) -> bool {
        self.genus >= 2
    }
    /// Add a known rational point.
    pub fn add_rational_point(&mut self, point: impl Into<String>) {
        self.known_rational_points.push(point.into());
    }
    /// Fermat curve x^n + y^n = 1 has genus (n-1)(n-2)/2 for n ≥ 3.
    pub fn fermat_curve(n: usize, field: impl Into<String>) -> Self {
        let genus = if n >= 3 { (n - 1) * (n - 2) / 2 } else { 0 };
        Self::for_curve(format!("x^{} + y^{} = 1", n, n), genus, field)
    }
}