User Duality

19 votes

1 answer

709 views

Discrepancy in Magma's calculation and Sage's of elliptic curve?

asked Jul 13, 2023 at 12:54

9 votes

1 answer

424 views

Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/\Bbb{Q}$ is bounded by some constant $M$, for all integers $D \in \Bbb{Z}$?

asked Oct 22, 2023 at 8:50

7 votes

1 answer

627 views

Cubic twist of elliptic curves and its rank

asked Dec 5, 2023 at 6:38

6 votes

1 answer

441 views

Ker of corestriction of Galois cohomology

asked Jun 17, 2023 at 11:05

6 votes

0 answers

543 views

Mistake in Silverman's book in proof of Neron-Ogg-Shafarevich criterion?

asked Sep 27, 2021 at 3:44

6 votes

1 answer

391 views

Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$

asked Feb 6, 2023 at 19:17

5 votes

2 answers

314 views

Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension

asked Dec 4, 2022 at 9:12

4 votes

1 answer

274 views

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

asked Mar 14, 2022 at 11:09

4 votes

1 answer

291 views

Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB

asked Jun 10, 2023 at 7:14

4 votes

1 answer

455 views

Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$

asked May 19 at 11:45

3 votes

1 answer

321 views

Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $

asked Sep 26, 2023 at 4:28

3 votes

0 answers

171 views

Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants

asked Nov 2, 2023 at 15:34

3 votes

2 answers

405 views

Infinitely many elliptic curve with twist rank more than $1$ in specific case

asked Jul 2, 2023 at 11:26

3 votes

1 answer

393 views

Completion of infinite degree extension of perfectoid fields is perfectoid?

asked Jun 5, 2022 at 5:35

3 votes

0 answers

220 views

Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?

asked Jul 16, 2022 at 7:33

3 votes

0 answers

110 views

Local global principle over infinite extension of $\Bbb{Q}$ which is not algebraically closed

asked Feb 21, 2023 at 11:02

3 votes

0 answers

127 views

Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$

asked Nov 20, 2022 at 8:23

2 votes

0 answers

140 views

$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group

asked May 8, 2023 at 18:24

2 votes

0 answers

147 views

Order $4$ element of Tate-Shafarevich group

asked May 23, 2023 at 17:14

2 votes

1 answer

173 views

How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$

asked Oct 5, 2022 at 22:21

2 votes

0 answers

190 views

Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$

asked Aug 2, 2022 at 15:56

2 votes

1 answer

285 views

Number of imaginary quadratic field with its ideal class group has $\Bbb{Z}/2\Bbb{Z}$ as 2 part

asked Jul 11, 2023 at 9:39

2 votes

0 answers

269 views

Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'

asked Feb 22 at 9:47

2 votes

1 answer

266 views

Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?

asked Nov 13 at 14:51

1 vote

1 answer

238 views

When $E_D:y^2=x^3+17D^2x$ has even rank?

asked Jul 12, 2023 at 18:00

1 vote

0 answers

119 views

Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$

asked Sep 20, 2023 at 14:04

1 vote

1 answer

347 views

Does $17x^4+y^2=-1$ have solution in $\Bbb{Q}_2(\sqrt{-5})$?

asked Jul 5, 2023 at 16:34

1 vote

0 answers

92 views

Inflation-restrction sequence for maximal $S$-ramified extension

asked Nov 16, 2023 at 6:25

1 vote

0 answers

100 views

Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant

asked Dec 27, 2023 at 17:53

1 vote

1 answer

118 views

Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality

asked Oct 13 at 14:51

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59 Questions

Discrepancy in Magma's calculation and Sage's of elliptic curve?

Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/\Bbb{Q}$ is bounded by some constant $M$, for all integers $D \in \Bbb{Z}$?

Cubic twist of elliptic curves and its rank

Ker of corestriction of Galois cohomology

Mistake in Silverman's book in proof of Neron-Ogg-Shafarevich criterion?

Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$

Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB

Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$

Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $

Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants

Infinitely many elliptic curve with twist rank more than $1$ in specific case

Completion of infinite degree extension of perfectoid fields is perfectoid?

Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?

Local global principle over infinite extension of $\Bbb{Q}$ which is not algebraically closed

Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$

$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group

Order $4$ element of Tate-Shafarevich group

How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$

Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$

Number of imaginary quadratic field with its ideal class group has $\Bbb{Z}/2\Bbb{Z}$ as 2 part

Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'

Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?

When $E_D:y^2=x^3+17D^2x$ has even rank?

Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$

Does $17x^4+y^2=-1$ have solution in $\Bbb{Q}_2(\sqrt{-5})$?

Inflation-restrction sequence for maximal $S$-ramified extension

Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant

Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality