Modular Curves and Quotients (Canonical Embeddings)

ModularCurveQuotient(N, A): RngIntElt, [RngIntElt] -> Crv

Raw   : BoolElt                    Default: false
Reduce: BoolElt                    Default: false

Given a level \(N\) and a (possibly empty) set of Atkin-Lehner involutions represented as a sequence of integers \(A\), this function computes a model for a quotient of \(X_0(N)\) by \(A\). When Raw is not set to true, an initial “semi-reduction” is not performed. If the Reduce option is true, then a complete LLL-reduction is performed on the resulting equations (this is impractical for genus greater than 50 or so). The returned curve can be: \(P^1\) for a genus 0 curve, an elliptic or hyperelliptic curve, or the canonical embedding of the curve in \(P^{g-1}\) where \(g\) is the genus of the quotient curve (in this general case, the coordinates correspond to cusp forms invariant under the specified Atkin-Lehner involutions).

X0NQuotient(N, A): RngIntElt, [RngIntElt] -> Crv

Raw   : BoolElt                    Default: false
Reduce: BoolElt                    Default: false

The same as ModularCurveQuotient, and the name used in the example below.

X0Nplus(N): RngIntElt -> Crv

The modular curve \(X_0(N)^+\), the quotient of \(X_0(N)\) by the Fricke involution \(w_N\). This is X0NQuotient(N, [N]).

X0Nstar(N): RngIntElt -> Crv

The modular curve \(X_0(N)^*\), the quotient of \(X_0(N)\) by the group generated by all of its Atkin-Lehner involutions. This is X0NQuotient(N, A) for A the sequence of prime powers \(p^{v_p(N)}\) over the primes \(p\) dividing \(N\).

GenusX0N(N): RngIntElt -> RngIntElt

The genus of \(X_0(N)\), computed from the standard formula rather than by constructing the curve.

GenusX1N(N): RngIntElt -> RngIntElt

The genus of \(X_1(N)\), computed from the standard formula rather than by constructing the curve.

GenusX0NQuotient(N, As): RngIntElt, [RngIntElt] -> RngIntElt

The genus of the quotient of \(X_0(N)\) by the group of automorphisms generated by the Atkin-Lehner involutions \(w_A\) for \(A\) in As, computed without constructing the quotient. Each \(A\) must satisfy \(A > 1\), \(A \mid N\) and \({\rm GCD}(A, N/A) = 1\).

AtkinLehnerNumberOfFixedPoints(N, A): RngIntElt, RngIntElt -> RngIntElt

The number of fixed points of the Atkin-Lehner involution \(w_A\) acting on \(X_0(N)\). Both \(N\) and \(A\) must be greater than \(1\), with \(A \mid N\) and \({\rm GCD}(A, N/A) = 1\).

Example: X0n Quotient (ex-5d6c55)

We compute a model for \(X_0(13\cdot 29)\) quotiented by the Atkin-Lehner involutions \(w_{13}\) and \(w_{29}\).

> C := X0NQuotient(13*29,[13,29]); C; // defined by cubics in P^4
Curve over Rational Field defined by
-x[1]*x[2]^2 + x[1]*x[2]*x[3] - x[2]^2*x[3] + x[1]*x[2]*x[4] +
    x[2]^2*x[4] + x[2]*x[4]^2 + x[1]*x[2]*x[5],
x[1]*x[2]^2 - x[2]^3 - x[2]^2*x[3] + x[1]*x[2]*x[5] + x[2]^2*x[5] +
    x[2]*x[4]*x[5],
x[1]*x[2]*x[3] - x[2]^2*x[3] - x[2]*x[3]^2 + x[1]*x[3]*x[5] +
    x[2]*x[3]*x[5] + x[3]*x[4]*x[5],
x[1]*x[2]*x[5] - x[2]^2*x[5] - x[2]*x[3]*x[5] + x[1]*x[5]^2 +
    x[2]*x[5]^2 + x[4]*x[5]^2,
-x[1]^2*x[2] + x[1]*x[2]^2 + x[1]*x[2]*x[3] - x[1]^2*x[5] -
    x[1]*x[2]*x[5] - x[1]*x[4]*x[5],
x[1]*x[2]*x[3] + x[2]^2*x[4] + x[2]*x[3]*x[4] + x[2]*x[4]*x[5] +
    x[3]*x[4]*x[5] + x[2]*x[5]^2 + x[4]*x[5]^2,
x[1]*x[2]*x[3] + x[1]*x[2]*x[4] + x[1]*x[2]*x[5] - x[1]*x[3]*x[5] -
    x[3]^2*x[5] + x[1]*x[4]*x[5] - x[2]*x[4]*x[5] - x[3]*x[4]*x[5] +
    x[1]*x[5]^2 + x[3]*x[5]^2,
-x[1]*x[2]*x[4] + x[1]*x[3]*x[4] - x[2]*x[3]*x[4] + x[1]*x[4]^2 +
    x[2]*x[4]^2 + x[4]^3 + x[1]*x[4]*x[5],
-x[1]*x[2]*x[3] + x[1]*x[3]^2 - x[2]*x[3]^2 + x[1]*x[3]*x[4] +
    x[2]*x[3]*x[4] + x[3]*x[4]^2 + x[1]*x[3]*x[5],
-x[1]*x[2]*x[5] + x[1]*x[3]*x[5] - x[2]*x[3]*x[5] + x[1]*x[4]*x[5] +
    x[2]*x[4]*x[5] + x[4]^2*x[5] + x[1]*x[5]^2,
x[1]*x[2]*x[4] - x[2]^2*x[4] - x[2]*x[3]*x[4] + x[1]*x[2]*x[5] -
    x[1]*x[3]*x[5] + x[2]*x[3]*x[5] - x[1]*x[5]^2,
-x[1]^2*x[3] - x[1]*x[2]*x[3] + x[1]*x[3]^2 - x[2]*x[3]^2 -
    x[1]*x[2]*x[4] - x[3]^2*x[4] - x[2]*x[4]^2 + x[1]*x[3]*x[5] -
    x[2]*x[4]*x[5] - x[4]^2*x[5] - x[1]*x[5]^2,
-x[1]^2*x[2] + x[1]*x[2]^2 + x[2]^2*x[3] + x[1]^2*x[4] -
    x[1]*x[3]*x[4] + x[2]*x[3]*x[4] + x[3]*x[4]^2 + x[1]^2*x[5] -
    x[1]*x[3]*x[5] - x[3]^2*x[5] - x[1]*x[4]*x[5] - x[3]*x[4]*x[5] +
    x[2]*x[5]^2 + x[3]*x[5]^2 + x[4]*x[5]^2,
-x[1]^2*x[2] + x[1]^2*x[3] - x[1]*x[2]*x[3] + x[1]^2*x[4] +
    x[1]*x[2]*x[4] + x[1]*x[4]^2 + x[1]^2*x[5],
x[1]^2*x[2] + x[1]^2*x[3] - x[1]*x[2]*x[3] + x[2]^2*x[3] - x[1]*x[3]^2
    - x[3]^3 - x[1]*x[2]*x[4] + x[1]*x[3]*x[4] - x[2]*x[3]*x[4] -
    x[3]^2*x[4] + x[1]^2*x[5] + x[1]*x[3]*x[5] - x[2]*x[3]*x[5] +
    x[3]^2*x[5] - x[3]*x[4]*x[5] + x[1]*x[5]^2
> Genus(C);
5

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