Mixing
2-Torsion in Jacobians of Curves Over Finite Fields
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Let $C$ be a (smooth, projective) curve over a finite field $mathbbF_q$, and let $J_C(mathbbF_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.
Question 1: Are there curves $C$ for which $J_C(mathbbF_q)$ is isomorphic, as a group, to $(mathbbZ/2mathbbZ)^k$ for some $k$? Can one characterize all such curves?
Question 2: What is known about upper bounds of the size of the $2$-torsion of $J_C(mathbbF_q)$ compared to the size of $J_C(mathbbF_q)$ itself? Is it necessarily true, say, that the 2-torsion is negligible if some parameter ($q$ or $g$) is large enough?
finite-fields jacobians
asked Aug 30 at 8:31
Ofir Gorodetsky
4,64611734
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up vote
6
down vote
favorite
Let $C$ be a (smooth, projective) curve over a finite field $mathbbF_q$, and let $J_C(mathbbF_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.
Question 1: Are there curves $C$ for which $J_C(mathbbF_q)$ is isomorphic, as a group, to $(mathbbZ/2mathbbZ)^k$ for some $k$? Can one characterize all such curves?
Question 2: What is known about upper bounds of the size of the $2$-torsion of $J_C(mathbbF_q)$ compared to the size of $J_C(mathbbF_q)$ itself? Is it necessarily true, say, that the 2-torsion is negligible if some parameter ($q$ or $g$) is large enough?
finite-fields jacobians
asked Aug 30 at 8:31
Ofir Gorodetsky
4,64611734
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Let $C$ be a (smooth, projective) curve over a finite field $mathbbF_q$, and let $J_C(mathbbF_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.
Question 1: Are there curves $C$ for which $J_C(mathbbF_q)$ is isomorphic, as a group, to $(mathbbZ/2mathbbZ)^k$ for some $k$? Can one characterize all such curves?
Question 2: What is known about upper bounds of the size of the $2$-torsion of $J_C(mathbbF_q)$ compared to the size of $J_C(mathbbF_q)$ itself? Is it necessarily true, say, that the 2-torsion is negligible if some parameter ($q$ or $g$) is large enough?
finite-fields jacobians
asked Aug 30 at 8:31
Ofir Gorodetsky
4,64611734
Let $C$ be a (smooth, projective) curve over a finite field $mathbbF_q$, and let $J_C(mathbbF_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.
Question 1: Are there curves $C$ for which $J_C(mathbbF_q)$ is isomorphic, as a group, to $(mathbbZ/2mathbbZ)^k$ for some $k$? Can one characterize all such curves?
Question 2: What is known about upper bounds of the size of the $2$-torsion of $J_C(mathbbF_q)$ compared to the size of $J_C(mathbbF_q)$ itself? Is it necessarily true, say, that the 2-torsion is negligible if some parameter ($q$ or $g$) is large enough?
finite-fields jacobians
asked Aug 30 at 8:31
Ofir Gorodetsky
4,64611734
asked Aug 30 at 8:31
Ofir Gorodetsky
4,64611734
asked Aug 30 at 8:31
Ofir Gorodetsky
4,64611734
asked Aug 30 at 8:31
Ofir Gorodetsky
4,64611734
4,64611734
add a comment |Â
add a comment |Â
1 Answer
1
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votes
up vote
9
down vote
accepted
I think $y^2=x^9-x$ over $mathbbF_3$ has $J_C(mathbbF_3)$ isomorphic to $(mathbbZ/2)^6$ but please check.
The $2$-torsion in $J_C$ over the algebraic closure is $(mathbbZ/2)^2g$ (or smaller in characteristic two). On the other hand, $#J_C(mathbbF_q) ge (sqrtq -1)^2g$, so for $q > 9$, the latter is bigger than the former (and usually much bigger). So the things you want can only exist for small $q$. I don't know a classification.
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
2
Thanks for making precise what immediately sprang to my mind, but only in a foggy, ill-formed way.
– Lubin
Aug 30 at 22:43
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
9
down vote
accepted
I think $y^2=x^9-x$ over $mathbbF_3$ has $J_C(mathbbF_3)$ isomorphic to $(mathbbZ/2)^6$ but please check.
The $2$-torsion in $J_C$ over the algebraic closure is $(mathbbZ/2)^2g$ (or smaller in characteristic two). On the other hand, $#J_C(mathbbF_q) ge (sqrtq -1)^2g$, so for $q > 9$, the latter is bigger than the former (and usually much bigger). So the things you want can only exist for small $q$. I don't know a classification.
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
2
Thanks for making precise what immediately sprang to my mind, but only in a foggy, ill-formed way.
– Lubin
Aug 30 at 22:43
add a comment |Â
up vote
9
down vote
accepted
I think $y^2=x^9-x$ over $mathbbF_3$ has $J_C(mathbbF_3)$ isomorphic to $(mathbbZ/2)^6$ but please check.
The $2$-torsion in $J_C$ over the algebraic closure is $(mathbbZ/2)^2g$ (or smaller in characteristic two). On the other hand, $#J_C(mathbbF_q) ge (sqrtq -1)^2g$, so for $q > 9$, the latter is bigger than the former (and usually much bigger). So the things you want can only exist for small $q$. I don't know a classification.
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
2
Thanks for making precise what immediately sprang to my mind, but only in a foggy, ill-formed way.
– Lubin
Aug 30 at 22:43
add a comment |Â
up vote
9
down vote
accepted
up vote
9
down vote
accepted
I think $y^2=x^9-x$ over $mathbbF_3$ has $J_C(mathbbF_3)$ isomorphic to $(mathbbZ/2)^6$ but please check.
The $2$-torsion in $J_C$ over the algebraic closure is $(mathbbZ/2)^2g$ (or smaller in characteristic two). On the other hand, $#J_C(mathbbF_q) ge (sqrtq -1)^2g$, so for $q > 9$, the latter is bigger than the former (and usually much bigger). So the things you want can only exist for small $q$. I don't know a classification.
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
I think $y^2=x^9-x$ over $mathbbF_3$ has $J_C(mathbbF_3)$ isomorphic to $(mathbbZ/2)^6$ but please check.
The $2$-torsion in $J_C$ over the algebraic closure is $(mathbbZ/2)^2g$ (or smaller in characteristic two). On the other hand, $#J_C(mathbbF_q) ge (sqrtq -1)^2g$, so for $q > 9$, the latter is bigger than the former (and usually much bigger). So the things you want can only exist for small $q$. I don't know a classification.
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
edited Aug 30 at 21:53
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
answered Aug 30 at 9:23
Felipe Voloch
26.7k563130
26.7k563130
2
Thanks for making precise what immediately sprang to my mind, but only in a foggy, ill-formed way.
– Lubin
Aug 30 at 22:43
add a comment |Â
2
Thanks for making precise what immediately sprang to my mind, but only in a foggy, ill-formed way.
– Lubin
Aug 30 at 22:43
2
2
Thanks for making precise what immediately sprang to my mind, but only in a foggy, ill-formed way.
– Lubin
Aug 30 at 22:43
Thanks for making precise what immediately sprang to my mind, but only in a foggy, ill-formed way.
– Lubin
Aug 30 at 22:43
add a comment |Â
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