All Questions
663 questions with no upvoted or accepted answers
1
vote
0
answers
129
views
How Galois group acts on Tate-Shafarevich group?
Let $L/K$ be a quadratic number field extension. Let $\operatorname{Sha}(E/L)$ be Tate-Shafarevich group of elliptic curve $E/L$.
How $\sigma \in \operatorname{Gal}(L/K)$ acts on $\operatorname{Sha}(E/...
- 1,541
1
vote
0
answers
187
views
Supersingular points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ and their Frobenius action
I am trying to understand certain points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ (where $p$ is a rational prime) via their moduli description.
A point $P \in X_0(p)$ can be viewed as $P:=(E, ...
- 637
1
vote
0
answers
150
views
Relative compactification without resolutions of singularities
Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
- 309
1
vote
0
answers
144
views
Characterization of Selmer group in terms of two descent
This question is about p 337 of Silverman's book ''The arithmetic of elliptic curves'', p 337, http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf.
Let $E:y^2=x^3+ax^2+bx$ and $E':Y^2=...
- 1,541
1
vote
0
answers
156
views
About the exact sequence $0\to $$Ш(E/\Bbb{Q})[\phi]\to Ш(E/\Bbb{Q})[2]\to Ш(E'/\Bbb{Q})[\hat{\phi}]$ in Silverman's book of elliptic curves
This question is about Silverman's book,$7$-th line from the bottom of $350$ page of 'The arithmetic of elliptic curves' (http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf) .
Let $E$ ...
- 1,541
1
vote
0
answers
237
views
Tate uniformization and reduction of elliptic curves
Let $E$ be an elliptic curve over $K$ (nonarchimedean) with $j$-invariant satisfying $|j(E)|>1$.
Tate uniformization theorem says that we have an isomorphism : $E \simeq \mathbf G_m/q^{\mathbf Z}$.
...
- 11
1
vote
0
answers
211
views
Period calculation for elliptic curve
In the paper "Hodge cycles on Abelian Varieties" (Proposition 1.5), Deligne proves the following theorem:
Let $X$ be a smooth projective variety over $\overline{\mathbf{Q}}$ of dimension $n$...
- 1,949
1
vote
0
answers
88
views
Pure, residual, 2-dimensional, semisimple $G_{\mathbb{Q}}$ representations
Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...
- 2,907
1
vote
0
answers
109
views
Why is the kernel cyclic if and only if the walk does not backtrack?
I'm reading Mathematics of Isogeny Based Cryptography by Luca De Feo. At some point (pg. 32), he says
"A walk of length $e_A$ in the $l_A$-isogeny graph corresponds to a kernel of size $l_A^{e_A};...
- 73
1
vote
0
answers
132
views
Is there an analog of Weil pairing for modular forms?
Given a newform $f(z)$ (of weight $k$) and a prime $p,$ consider the classical Galois representation
$$\rho_{f,p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}/p\mathbb{Z}).$...
- 521
1
vote
0
answers
180
views
How to link the rank of Elliptic Surfaces and Elliptic Curves over function fields?
I have been investigating certain elliptic surfaces for my research, and when giving a presentation, I was asked why when given an elliptic surface $E$, say given in Weierstrass form
$$E\colon y^2+a_1(...
- 11
1
vote
0
answers
95
views
An elliptic threefold and the Mordell–Weil lattices of its reductions
Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\...
- 1,833
1
vote
0
answers
144
views
Elliptic curves whose $2,3,5$-parts of Sha are large
Let $E$ be an elliptic curve, and $\text{Sha}(E)$ its Shafarevich-Tate group which measures the failure of the local-to-global principle for the curve. It is conjectured that $\text{Sha}(E)$ is a ...
- 27k
1
vote
0
answers
180
views
Homomorphism of formal group of elliptic curve corresponding to its endomorphism
Let $E$ be an elliptic curve and $ \hat{E}$ be its formal group.
Rubin's lemma $3.7$ in 'Elliptic curves with complex multiplication' reads
For arbitrary $φ∈End(E)$, there exists unique $φ(t)∈End( \...
- 1,541
1
vote
0
answers
76
views
How to construct explicitly defining polynomials of an morphism between smooth irreducible curves?
Let $\phi\!: C_1 \to C_2$ be a separable morphism of smooth irreducible curves embedded as projectively normal models by invertible sheaves $\mathcal{L}_1$ and $\mathcal{L}_2$ respectively. Theorem 4....
- 1,833
1
vote
0
answers
63
views
Parametrizations of elliptic curves that "fixes" torsion points
I am not entirely sure this question is research-level question, but I have tried stack exchange and have received no response there.
During a conversation with a professor, I was informed of the ...
- 101
1
vote
0
answers
100
views
Iterated integrals on higher dimensional Calabi-Yau manifolds?
I recently read about the construction of closed quasi-periodic differential forms on elliptic curves (1-dim Calabi-Yaus) via the Kronecker-Eisenstein series. I now wonder if similar constructions are ...
- 231
1
vote
0
answers
112
views
Isomorphism of Brauer groups of curves
I asked this question a few days ago on math.stackexchange with no success, and it doesn't seem like there'll be any. So I thought I'll repost it here.
A recent big result proved by $\mathrm{\check{C}}...
- 895
1
vote
0
answers
88
views
The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions
Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map
\begin{equation}
\mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
- 437
1
vote
0
answers
275
views
4-distance problem and elliptic curves
The 4-distance problem is an open question(as far as I know it is still open) that asks if there exists a point P on the Euclidean plane such that its distances to all four points of a unit square are ...
- 547
1
vote
0
answers
58
views
A lower bound for a sum related to the $j$-invariant function
There are some days that I am thinking in the following problem.
For any positive integer $x$, let $t(x)$ be a real number which a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \...
- 515
1
vote
0
answers
126
views
Relation between $L$-values of elliptic curves and Manin constants
Given an elliptic curve $E$ over $\mathbf{Q}$, we can attach two numbers two it.
the so-called Manin constant $c_E$. (Defined below the fold.)
the "algebraic $L$-value" given by $L(E,1)/\...
- 1,949
1
vote
0
answers
246
views
To justify the intuition about #$E(\Bbb Q_p)$＝$∞$
Let $E$ be an elliptic curve on $\Bbb Q_p$.
$E_0(\Bbb Q_p)$ is points of $E(\Bbb Q_p)$ reduced to nonsingular points.
How to prove #$E(\Bbb Q_p)$＝$∞$ directly ?
According to Silverman's book 'the ...
- 1,541
1
vote
0
answers
208
views
Outline of the proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite
I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite.
Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ?
My ...
- 1,541
1
vote
0
answers
167
views
The existence of two $p$-isogenies implies the existence of one $p^2$-cyclic isogeny
Let $E$ be an elliptic curve over $\mathbb{Q}$.
(or over a number field $K$.)
If $E$ has two $p$-isogenies over $\mathbb{Q}$, then $E$ has $p^2$ cyclic isogeny over $\mathbb{Q}$.
I want to show it ...
- 185
1
vote
0
answers
194
views
Brauer-Manin obstruction and affine curves
I'm looking for references that can justify to what extent is the following statement true:
Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...
- 895
1
vote
0
answers
143
views
A specific Diophantine equation related to the congruent number question
Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
- 159
1
vote
0
answers
191
views
Showing that two families of elliptic curves are diffeomorphic
Consider a family of elliptic curves over the open unit disc $D\subset \mathbb{C}$ which degenerates to the nodal elliptic curve over the point $0$. I'd like to show that such a family is ...
- 191
1
vote
0
answers
78
views
Elliptic fibrations on some Kummer surface in characteristic $2$
In the question I ask about one elliptic fibration on the surface
$$
K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2.
$$
over a finite field $\mathbb{F}_q$ of characteristic $2$ such ...
- 1,833
1
vote
0
answers
104
views
A subgroup of the $n$-Selmer group
Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$.
Let $E$ be an elliptic curve over a number field $F$.
The $n$-Selmer group, denoted by $S_n(...
- 1,283
1
vote
0
answers
190
views
Monodromy Representation on $H_1$ of Elliptic Curve
I'm reading this post by Charles Siegel on Monodromy Representations
and there is a construction in example a not unterstand.
We look at the family $y^2z=x(x-z)(x-\lambda z)$ of projective elliptic
...
- 6,006
1
vote
0
answers
109
views
how often can a fixed prime be anomalous?
Let $p$ be a fixed prime. Say for simplicity $p>5$. As we vary over all elliptic curves $E/\mathbb{Q}$ of height $< X$, can one (expect to) say anything about what proportion of elliptic curves ...
- 1,283
1
vote
0
answers
51
views
Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
1
vote
0
answers
562
views
Efficient deterministic algorithm for quadratic residues
In Elliptic Cryptography, one may need the calculation of the quadratic residue over a finite field to generate a random point on the curve.
In simple terms, given the equation of the curve as short ...
- 115
1
vote
0
answers
134
views
Automorphic representation of weight 3 eigenforms
Let $f$ be a weight 3 eigenform with rational Fourier coefficients. As shown by Elkies and Schutt, $f$ is associated to a singular K3 surface over $\mathbb{Q}$. A construction of Shioda and Inose ...
- 167
1
vote
0
answers
134
views
Does the Galois groups $G_1$ and $G_2$ are isomorphic under `some suitable assumptions`?
Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.
Consider the $p$-power torsion points and adjoin them with $K$.
Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-...
- 930
1
vote
0
answers
147
views
Drinfeld basis of the universal formal deformation of a supersingular elliptic curve
Now I'm trying (5.3.2) of Katz-Mazur's Arithmetic moduli of elliptic curves.
Let $k$ be an algebraically closed field of characteristic $p > 0$, $E_0$ a supersingular elliptic curve over $k$, ...
- 1,364
1
vote
0
answers
94
views
Is there an infinite order $\mathbb{F}_{p}$-section for a certain elliptic surface $\mathcal{E}_n$?
Consider a natural number $n$, a finite field $\mathbb{F}_{p}$ (such that $p$ is prime, $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, and $\sqrt[3]{2} \notin \mathbb{F}_p$), ...
- 1,833
1
vote
0
answers
504
views
The Picard scheme of an ordinary singular curve
Let $k$ be an algebraically closed field, $C$ a proper reduced connected scheme over $k$ of dimension 1, whose singularity is at worse ordinary, $\pi : \tilde{C} \to C$ the normalization of $C$ and $...
- 1,364
1
vote
0
answers
77
views
Sato-Tate for length zero intervals
Let $E$ be an elliptic curve (without CM) over a number field $K.$ Is it known that $a(\mathfrak{p})=N(\mathfrak{p})+1-|E(\mathbb{F}_{\mathfrak{p}})|$ is neither zero, not $2\sqrt{N(\mathfrak{p})}$ ...
- 11
1
vote
0
answers
212
views
Generate algorithmically an elliptic curve with its exact class group structure?
Is it possible to generate an elliptic curve $E$ (randomly), together with knowing its class group $\mathrm{Cl}(\mathcal{O})$ structure? where $\mathcal{O}$ is its endomorphism rings $\mathsf{End}(E)$ ...
- 181
1
vote
0
answers
325
views
Compatiblity of completion and fibre products. (Formal completion and formal groups)
Let $S$ be a scheme (not necessarily locally noetherian), $X$ a smooth separated group scheme over $S$, and $\hat{X}$ be the formal completion along with the identity section.
Then does the group ...
- 1,364
1
vote
0
answers
234
views
Why is the $\mathbb{Z}_p$-corank of $\operatorname{Sel}_{p^\infty}(E/\mathbb{Q})$ finite?
I'm interested on the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$. I read that the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group $\operatorname{Sel}_{p^\infty}(E)\doteq\...
user105226
1
vote
0
answers
83
views
Can someone suggest some references for rank discussion of elliptic curves (bonus if it systematic)?
First, I apologise if such a question has been asked before. Please feel free to refer me to the previous question, if it answers my current query then I will delete this post.
I am reading the ...
- 401
1
vote
0
answers
220
views
Shortest possible reasonably self-contained formulation of the modularity theorem
This is question in mathematical exposition, not research, I hope this is ok.
I am writing a book about great theorems. My question is: what is the shortest formulation of the modularity theorem, ...
- 7,182
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
1
vote
0
answers
148
views
Lang's proof of the Shimura's exact sequence
Let $E$ be an elliptic curve having invariant $j$ and defined over $\mathbf C(j)$. Let $\sigma$ be an automorphism of $F_{\mathbf C}$, the field of all modular functions of all levels. Let $p$ be a ...
- 2,375
1
vote
0
answers
95
views
Zeros of modular functions and automorphisms
Let $F_N$ be the field of modular functions of level $N$ and with Fourier coefficients in $\mathbf Q(\zeta_N)$. We have
$$F_N=\mathbf Q(j, f^{(r,s)}_N),$$
where the $f^{(r,s)}_N$ are the Fricke ...
- 2,375
1
vote
0
answers
68
views
Non-vanishing Taylor coefficients and Poincaré series
I'm studying these algebraic numbers in the table below and have succesfully shown what i wanted with the given values I had.
The table is found in the book "The 1-2-3 of Modular Forms" by ...
1
vote
0
answers
75
views
Is there a non-singular irreducible genus $2$ curve $C/\mathbb{F}_p$ and two $\mathbb{F}_p$-coverings $C \to E$, $C \to E^{(1)}$ of some degree $n$?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 = ...
- 1,833