All Questions
663 questions with no upvoted or accepted answers
2
votes
0
answers
286
views
Moduli interpretation and Ogg's notation for the cusps on modular curves
In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps,
that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
- 1,364
2
votes
0
answers
198
views
Finding rational points via birational map
Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$
and let $\overline{C}$ denote the projective closure of $C$. For ...
- 21
2
votes
0
answers
601
views
Summation form of the Hasse-Weil zeta function?
The only way I know to write the Hasse-Weil zeta function of an elliptic curve is as a product over the local zeta factors which are rational functions. To me, this appears like an Euler product.
Is ...
- 3,730
2
votes
0
answers
210
views
What is the minimal model of $E:y^2=x^3-x-n$?
Does it hold that $E:y^2=x^3-x-n$ is a minimal model for any choice $n$? Using the Sage programming language we can check that $E$ is indeed minimal for every $n\leq20,000$.
- 2,902
2
votes
0
answers
309
views
Merel's theorem on uniform bound for torsion of all elliptic curves
I am reading Silverman's book on Arithmetic of elliptic curves. There he mentions a theorem of Merel (Thm 7.5.1) which reads like this.
Thm: For every integer $d \geq 1$, there is a constant $N(d)$ ...
- 313
2
votes
0
answers
78
views
What are possible applications of 'fast arithmetic' in the Jacobian (degree zero Picard group) of projective curves over fields?
It is well known that there are plenty possible applications of 'fast arithmetic' (that is, 1. having an algorithm at hand that actually computes in..., and 2. the running time of that algorithm is ...
- 435
2
votes
0
answers
228
views
Tate module of elliptic curves; Commuting Hom functor and tensor product in the second coordinate
Let $\Lambda$ be the Iwasawa Algebra of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_{cyc}$ of $\mathbb{Q}$. Let $\widehat{\Lambda}$ be its Pontryagin dual (i.e the ...
- 313
2
votes
0
answers
155
views
Reference for the $3$-series of an elliptic formal group law
The $3$-series of the formal group law of the Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x$ begins
$$
[3](z) = 3 z - 8 a_2 z^3 + (24 a_2^2 - 96 a_4) z^5 - (72 a_2^3 - 288 a_2 a_4) z^7 + (216 a_2^4 - ...
- 9,263
2
votes
0
answers
102
views
Rankin-Selberg method and Symmetric power of elliptic curves
Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$. Let $f=\sum a_n q^n$ be the weight 2 modular form corresponding to $E$.
Define $L_2(f,s)=\zeta(s-1)L(Sym^2(E),s)$.
The following ...
- 609
2
votes
0
answers
300
views
Elliptic curves: about a passage in J. Silverman's "Advanced topics of elliptic curves"
Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage ...
- 675
2
votes
0
answers
100
views
Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
2
votes
0
answers
101
views
$\mathscr{M}_*$, the stack of generalized elliptic curves (with some additional conditions) is locally of finite presentaion
Let $\mathscr{M}_*$ be the stack over $\mathbb{Z}$ which classifies generalized elliptic curves $E/S$, such that for every geometric point $k$ of $\mathbb{Z}$, the fibre $E_k$ is smooth or $n$-gon, ...
- 1,364
2
votes
0
answers
95
views
Can the cohomology group $H^1(H, E[p])$ be trivial for all subgroups $H$ of $Gal(K(E[p])/K) \simeq GL_2(\mathbb Z/p)$?
Let $p$ be a fixed odd prime, $K$ be a number field and $E$ be an elliptic curve defined over $K$. Set $L =K(E[p])$.
We know that $G=Gal(L/K)$ is a subgroup of $GL_2(\mathbb Z/p)$.
Suppose $G =GL_2(\...
- 1,283
2
votes
0
answers
65
views
What conditions are sufficient for two points to be independent in the Mordell-Weil group?
Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface
$$
\mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6(t),
$$
where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly ...
- 1,833
2
votes
0
answers
90
views
Cubic extensions of number fields and their local nature
Let $F$ be an irreducible squarefree cubic polynomial over a number field $K$. Let $L:=K[x]/{(F(x))}$ be a cubic extension of $K$.
Suppose that $\alpha \in L^\times$ such that $N_{L/K}(\alpha)=\beta^...
- 335
2
votes
0
answers
243
views
Some questions regarding computation of the Mordell-Weil group
I was reading the theory relevant to Selmer and Shafarevich-Tate groups from Silverman's AEC. And I have a lot of doubts related to these topics:
First, I don't understand the reasoning behind the ...
- 401
2
votes
0
answers
206
views
How often a number can be conductor of an elliptic curve
There are several upper bounds for number of elliptic curves (over Q, say) upto-isomorphism with a given conductor N. Probably the best one is given by Helfgott-Venkatesh of order N^{0.22} (or may be ...
- 521
2
votes
0
answers
58
views
Number of representations by the norm in a division algebra corresponding to endomorphism rings of elliptic curves
Let $E$ be a supersingular curve over a field of characteristic $p$ with endomorphism ring $\mathcal O_D$ which is a maximal order in a division ring $D$ over $Q$ ramified at $p$ and $\infty$.
The ...
- 7,746
2
votes
0
answers
807
views
Why is the congruent number problem open?
I was reading up about the congruent number problem.
One of the theorems on the subject says how the two things are equivalent: a positive integer $n$ being a congruent number and elliptic curve $y^...
- 401
2
votes
0
answers
131
views
The field generated by the torsion points of an elliptic curve
Let $E$ be an elliptic curve with complex multiplication by an order $\mathcal O$ in an imaginary quadratic field $K$. Let $H=K(j(E))$ and $$L_N=K(j(E),E[N])=H(E[N]).$$
It is not hard to prove that
$...
- 2,375
2
votes
0
answers
177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142
2
votes
0
answers
541
views
Trace of Frobenius on $p$-adic Tate module
Let $k$ be a finite field of characteristic $p>0$ with $q$ elements. Denote $K=W(k)[1/p]$.
Let $E$ be an elliptic curve over $W(k)$ with good reduction.
Choose a lifting $\mathrm{Frob} \in \...
- 429
2
votes
0
answers
101
views
Some Open Problems from a Paper on Isomorphism Classes of Elliptic Curves over a Finite Field?
I am reading the following article on isomorphism classes of elliptic curves over a finite field https://pdfs.semanticscholar.org/7280/b7c66cf02f1d43cbe042d7a6f6b4b7de269c.pdf and noticed some open ...
- 5,146
2
votes
0
answers
241
views
Rational curves on ruled surfaces
Let $S$ be a ruled surface (over an algebraically closed field) with an $\mathbb{P}^1$-bundle $\pi\!: S \to E$ onto an elliptic curve $E$. What is the classification of (possibly singular) irreducible ...
- 1,833
2
votes
0
answers
85
views
Finding Coefficients of a Pairing Friendly Elliptic Curve
Assuming that I know the size of the base field ($q$), size of the prime order subgroup ($r$), and embedding degree of the curve ($k$) of a pairing friendly elliptic curve ($E$), I would like to ...
- 121
2
votes
0
answers
114
views
Algebraic theta-functions of level $2$ on an elliptic curve
Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $...
- 1,833
2
votes
0
answers
60
views
A conjectural formula for the "minimal degree function", $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$
THE RECURSION: $f\rightarrow A(f)$
$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^...
- 5,422
2
votes
0
answers
171
views
trivial solutions for Diophantine equations
Let $K$ be an odd degree number field. Consider the Diophantine equation:
$$
X^4 + bY^4 =Z^2
$$
where $b\neq 0$.
Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
- 1,283
2
votes
0
answers
162
views
Notation for endomorphism algebra of Elliptic Curves
$\newcommand{\End}{\operatorname{End}}$For an elliptic curve $E$, I understand that the notation $\End(E)$ denotes the ring of endomorphisms of $E$. Since $\End(E)$ is torsion free, it's possible to ...
2
votes
0
answers
94
views
The quotient of a superspecial abelian surface by the involution
Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
- 1,833
2
votes
0
answers
290
views
Elliptic fibrations on the Fermat quartic surface
Consider the Fermat quartic surface
$$
x^4 + y^4 + z^4 + t^4 = 0
$$
over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$).
Is there the full ...
- 1,833
2
votes
0
answers
234
views
Global minimal Weierstrass equation over function fields
Let $K$ be a field of characteristic zero, $V/K$ a smooth projective variety and $F=\overline{K}(V)$ be the function field of $V$ over $\overline{K}$. For any elliptic curve $E/F$, it has a Weiertrass ...
- 21
2
votes
0
answers
182
views
Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)
Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...
- 3,625
2
votes
0
answers
304
views
Surjectivity of map of Picard schemes implies abelian
Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here.
I am looking for a reference or explanation of the fact that is used in Mumford'...
- 203
2
votes
0
answers
121
views
Global invariant cycles in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
- 728
2
votes
0
answers
180
views
Two elliptic curves not dominated by a genus two curve
Let $k$ be a number field and let $E$ and $E'$ be elliptic curve over $k$.
There is a genus two curve $X$ over $\overline{k}$ which dominates $E$ and $E'$.
Question. Is there a genus two curve $X$ ...
- 21
2
votes
0
answers
365
views
Formula for multiplication by $n$ on an elliptic curve
Let $E$ be an elliptic curve, given by a Weierstrass equation $y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$.
If $P = (x, y)$ is a point on $E$ and $n$ is a (positive) integer, then the point $...
- 3,432
2
votes
0
answers
222
views
Exact ramification information of mod $p$ Galois representation
Let $E$ be an elliptic curve over a number field $F$. As usual, let
$\bar{\rho}:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}(E[p])$ be the mod $p$ Galois representation associated to $E$. It is known ...
- 1,428
2
votes
0
answers
195
views
Specialization Theorem for function fields
I just want to know that whether the specialization theorem holds for multi variable function fields or not? i mean when we specialize over one function field to the other , can be sure that Mordell-...
- 131
2
votes
0
answers
719
views
Self intersection of theta divisor
I hope my question is not too basic here.
I have a very specific setting and I am trying to not use "big theorems" to prove a well known fact algebraically. I hope someone can give me a hint.
Let $J/...
2
votes
0
answers
286
views
Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
- 85
2
votes
0
answers
228
views
p-divisible groups over a p-adic field
p-div gps over a finite field (e.g., $\mathbb F_p$) are classified by Dieudonne modules.
There are also results for p-div gps over integers of local fields (e.g., over $\mathbb Z_p$).
However, are ...
- 153
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
148
views
Purely inseparable $k$-rational dominant maps between an absolutely irreducible $k$-surface and $\mathbb{P}^2$
Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?
There is a purely ...
- 1,833
2
votes
0
answers
87
views
Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension
If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:
(1.) $\...
- 1,805
2
votes
0
answers
336
views
Reduction "modulo $p$" of $\mathfrak{p}$-torsion points of CM elliptic curves
Let $E/L$ be an elliptic curve defined over a number field $L$. Assume moreover that $E$ has complex multiplication by an imaginary quadratic field $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. ...
- 7,609
2
votes
0
answers
294
views
"Algebrazing" canonical subgroups of elliptic curves
I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...
- 845
2
votes
0
answers
178
views
what is the structure of the group of isogenies between two ordinary elliptic curve?
Let $E_1$ and $E_2$ be two ordinary elliptic curves. It is well known that the group $Home(E_1,E_2)$ is a $\mathbb{Z}$ module of rank at most four. In the case where $E_1=E_2$, this module has rank ...
- 369
2
votes
0
answers
476
views
Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?
Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
- 16.9k
2
votes
0
answers
94
views
Complex Structure Moduli of Elliptic Fibrations
Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...