Skip to main content
MathOverflow

Return to Answer

added 4 characters in body
Source Link
Francesco Polizzi
Francesco Polizzi
  • 66.3k
  • 5
  • 180
  • 283

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that, for any $t \in \mathbb{N}$, the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exist elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. Setting $$N_{A}(t):= \{\textrm{number of elliptic curves on A such that} \deg _{\mathcal{L}} C \leq t \},$$ some asymptotic upper bounds for $N_A(t)$ are available. In fact, computing $N_A(t)$ is the same as finding the number of solutions of certain diophantine equations.

For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007.

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that for any $t \in \mathbb{N}$ the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exist elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. Setting $$N_{A}(t):= \{\textrm{number of elliptic curves on A such that} \deg _{\mathcal{L}} C \leq t \},$$ some asymptotic upper bounds for $N_A(t)$ are available. In fact, computing $N_A(t)$ is the same as finding the number of solutions of certain diophantine equations.

For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007.

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that, for any $t \in \mathbb{N}$, the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exist elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. Setting $$N_{A}(t):= \{\textrm{number of elliptic curves on A such that} \deg _{\mathcal{L}} C \leq t \},$$ some asymptotic upper bounds for $N_A(t)$ are available. In fact, computing $N_A(t)$ is the same as finding the number of solutions of certain diophantine equations.

For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007.

added 275 characters in body
Source Link
Francesco Polizzi
Francesco Polizzi
  • 66.3k
  • 5
  • 180
  • 283

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that for any $t \in \mathbb{N}$ the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there existsexist elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. ForSetting $$N_{A}(t):= \{\textrm{number of elliptic curves on A such that} \deg _{\mathcal{L}} C \leq t \},$$ some asymptotic upper bounds for $N_A(t)$ are available. In fact, computing $N_A(t)$ is the same as finding the number of solutions of certain diophantine equations.

For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007  .

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that for any $t \in \mathbb{N}$ the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exists elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007  .

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that for any $t \in \mathbb{N}$ the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exist elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. Setting $$N_{A}(t):= \{\textrm{number of elliptic curves on A such that} \deg _{\mathcal{L}} C \leq t \},$$ some asymptotic upper bounds for $N_A(t)$ are available. In fact, computing $N_A(t)$ is the same as finding the number of solutions of certain diophantine equations.

For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007.

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

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that for any $t \in \mathbb{N}$ the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exists elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007 .