Skip to main content
MathOverflow

Return to Answer

added 10 characters in body
Source Link
Francesco Polizzi
Francesco Polizzi
  • 66.3k
  • 5
  • 180
  • 283

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, partially answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?"

The result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?"

The result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, partially answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?"

The result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

added 5 characters in body
Source Link
Francesco Polizzi
Francesco Polizzi
  • 66.3k
  • 5
  • 180
  • 283

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?"

The resultsresult is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17 answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?

The results is the following:

THEOREM Consider a product of elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?"

The result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

Source Link
Francesco Polizzi
Francesco Polizzi
  • 66.3k
  • 5
  • 180
  • 283

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17 answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?

The results is the following:

THEOREM Consider a product of elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.