All Questions
125
questions
2
votes
1
answer
295
views
Why this genus one curve over $\mathbb{F}_5$ appear to violate Hasse-Weil bound?
Working over $\mathbb{F}_5$, the affine curve $x^4+2=y^2$ has no points.
The projective curve $x^4+2y^4=z^2y^2$ has only one point $(0 : 0 : 1)$.
Both curves appear to violate Hasse-Weil bound of $4....
- 24.7k
8
votes
0
answers
290
views
Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 821
1
vote
0
answers
99
views
Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12k
4
votes
1
answer
333
views
Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$
Let $E/K$ be an elliptic curve over number field $K$. Let $M_K$ be a set of all places of $K$.
My question is, Does there exist a finite set $S\subset M_K$ such that
$\forall C$: $E/K$-torsor, $\...
- 1,467
3
votes
0
answers
146
views
Examples of curves $C$ with $\operatorname{Jac}(C) \cong E^3$, $E$ a CM elliptic curve
Let $k$ be a field of your choice— I'm particularly interested in algebraically closed fields. Are there explicit examples of curves over $k$ whose Jacobian is isogenous to the product of three copies ...
- 371
0
votes
0
answers
77
views
Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra
Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1.
Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
- 538
0
votes
1
answer
336
views
Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$
$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$.
Let $\sigma$ be a generator of $\Gal(L/K)$.
Let $E/K$ be an elliptic curve defined ...
- 1,467
0
votes
1
answer
260
views
Is there an isotrivial elliptic surface of positive rank having a section of order $3$?
Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil ...
- 1,813
2
votes
0
answers
129
views
Isom-functor for generalized elliptic curves is representable
I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,
(page DeRa-61) (*) For $C_i$, ...
- 21
3
votes
0
answers
201
views
Tate isogeny theorem over varieties?
Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
- 341
4
votes
1
answer
171
views
Primes of bad reductions for quotients of elliptic curves
Let $E$ be an elliptic curve over a number field $K$ and $p$ a prime. Suppose that $E$ has a $K$-rational $p$-torsion, which gives the short exact sequence $0\to\mathbb{Z}/p\to E[p]\to\mu_p\to0$ of ...
- 1,408
1
vote
1
answer
212
views
When $E_D:y^2=x^3+17D^2x$ has even rank?
Let $E:y^2=x^3+17x$ be an elliptic curve.
In this MO page(Infinitely many elliptic curve with twist rank more than $1$ in specific case), Nulhomologous's and other's comment reads from parity ...
- 1,467
3
votes
2
answers
346
views
Infinitely many elliptic curve with twist rank more than $1$ in specific case
Let $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer.
It is conjectured that 50％ of twist of elliptic curve $E_D$ has rank $0$ and $50％$ has rank $1$.
But is some particular ...
- 1,467
0
votes
0
answers
148
views
Norm map of Tate-Shafarevich group $\mathrm{Sha}(E/K)\to \mathrm{Sha}(E/\Bbb{Q})$
Let $K$ be a quadratic number field. Let $\sigma$ be a generator of Galois group of $K/\Bbb{Q}$. Let $E$ be an elliptic curve defined over $\mathbb{Q}$.
Let $\mathrm{Sha}(E/K)$ denote the Tate-...
- 1,467
4
votes
1
answer
271
views
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
The result of LMFDB claims (https://www.lmfdb.org/EllipticCurve/Q/1640/c/1 )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. ...
- 1,467