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ID: INPUT{id.yaml}

Title: >
  Equations satisfied by Igusa invariants of split Jacobians

Definition: >
  This table contains equations satisfied by the 
  Igusa Invariants of genus $2$ curves with $(n,n)$ split Jacobians.
  More precisely, 
  let $C$ be a genus $2$ curve, 
  $Y^2 = f(X)$ a model thereof, 
  $[I_2:I_4:I_6:I_8:I_{10}]$ be the Igusa invariants of this curve,
  and $(i_1,i_2,i_3)=(144 I_4/I_2^2,-1728(I_2 I_4-3 I_6)/I_2^3,486 I_{10}/I_2^5)$ 
  the absolute invariants.
  If $\text{Jac}(C)$ is optimally $(n,n)$-split then its absolute invariants 
  satisfy the equation $\phi_n=0$ with $\phi_n$ in this table.
  
Parameters:
  n:
    display: $n$
    type: Z
    constraints: $n \geq 2$
   
Comments:
  comment-more-data: >
    More complete data is available at CITE{BrDoWeb}.
  
  comment-sources: >
    The case $n=2$ is classical, 
    $n=3$ due to Kuhn CITE{Kuhn}, and
    $n=4$ due to Bruin and Doerksen CITE{BrDo} (not listed in this table due to the size, available at CITE{BrDoWeb}).
  
Formulas:

Programs:

References:
  Bol:
    bib: >
      Oskar Bolza, "Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades", 
      Math. Ann. 28 (1887), no. 3, 447-456. 
  BrDo:
    bib: >
      Nils Bruin, Kevin Doerksen, 
      "The arithmetic of genus two curves with (4,4)-split Jacobians",
      Canad. J. Math. 63 (2011), 992-1021.
    arXiv: 0902.3480
  Kuhn:
    bib: >
      R. M. Kuhn, "Curves of genus 2 with split Jacobian".
      Trans. Amer. Math. Soc. 307(1988), no. 1, 41-49.

Links:
  BrDoWeb:
    title: "N. Bruin, K. Doerksen: Equations satisfied by Igusa Invariants of split Jacobians" 
    url: http://www.cecm.sfu.ca/~nbruin/splitigusa/
  LMFDB-Igusa:
    title: "LMFDB knowledge: Igusa invariants of a genus 2 curve"
    url: https://www.lmfdb.org/knowledge/show/g2c.igusa_invariants

Similar tables:
  
Keywords:
  
Tags:
- number theory
- curves

Data properties:
  type: Z[]
  complete: no
  sources:
  - CITE{BrDoWeb}
 
Display properties:
  number-header: $\phi_n$

Data: INPUT{polynomials.yaml}

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Equations satisfied by Igusa invariants of split Jacobians

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Polynomials

$n$

$\phi_n$

INPUT{polynomials.yaml} (not shown in preview)

Definition

This table contains equations satisfied by the Igusa Invariants of genus $2$ curves with $(n,n)$ split Jacobians. More precisely, let $C$ be a genus $2$ curve, $Y^2 = f(X)$ a model thereof, $[I_2:I_4:I_6:I_8:I_{10}]$ be the Igusa invariants of this curve, and $(i_1,i_2,i_3)=(144 I_4/I_2^2,-1728(I_2 I_4-3 I_6)/I_2^3,486 I_{10}/I_2^5)$ the absolute invariants. If $\text{Jac}(C)$ is optimally $(n,n)$-split then its absolute invariants satisfy the equation $\phi_n=0$ with $\phi_n$ in this table.

Parameters

$n$

— integer ($n \geq 2$)

Comments

(1)

More complete data is available at [4].

(2)

The case $n=2$ is classical, $n=3$ due to Kuhn [3], and $n=4$ due to Bruin and Doerksen [2] (not listed in this table due to the size, available at [4]).

References

[1]

Oskar Bolza, "Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades", Math. Ann. 28 (1887), no. 3, 447-456.

[2]

Nils Bruin, Kevin Doerksen, "The arithmetic of genus two curves with (4,4)-split Jacobians", Canad. J. Math. 63 (2011), 992-1021. (arXiv)

[3]

R. M. Kuhn, "Curves of genus 2 with split Jacobian". Trans. Amer. Math. Soc. 307(1988), no. 1, 41-49.

Links

[4]

N. Bruin, K. Doerksen: Equations satisfied by Igusa Invariants of split Jacobians

[5]

LMFDB knowledge: Igusa invariants of a genus 2 curve

Data properties

Entries are of type: integral polynomial

Table is complete: no

Sources of data: [4]