All Questions
1,978 questions
3
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p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 12.9k
0
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0
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58
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Given $N=pq$, with what complexity we can find integer $a : q-\sqrt{q} < a <q+\sqrt{q}$?
Related to open problem and this question.
Let $N=p q$ be integer with unknown factorization.
Q1 Given $N=pq$, with what complexity we can find integer $a : q-\sqrt{q} < a <q+\sqrt{q}$?
Assume ...
- 25.4k
3
votes
0
answers
36
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Avoiding class/unit group computation when computing $p$-Selmer groups
Let $K$ be a number field, $S$ be a finite set of places of $K$, and $K(S,p)$ be the $p$-Selmer group of the $S$-integers of $K$, that is the set of nonzero elements of $K$ modulo $p$-th powers whose ...
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
9
votes
3
answers
737
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Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
- 1,597
5
votes
1
answer
204
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Isogenous elliptic curves and canonical modular polynomials
Let $\ell$ and $p$ be two primes. We are looking for a method for checking whether two supersingular elliptic curves over the finite field $F_p$, given through their $j$-invariants, are $\ell$-...
CommunityBot
- 1
0
votes
0
answers
132
views
How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
1
vote
1
answer
121
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Rational functions on elliptic curves over global fields with given support
Let $E$ be an elliptic curve over a global field $k$. Let $x_1, \dots, x_r$ be a set of generators of $E(k) / E(k)_{tor}$ (or more generally, a $\mathbb Q$-basis of $E(k)_{\mathbb Q}$), and let $x_0$ ...
- 105k
2
votes
0
answers
110
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How to apply Pohlig Hellman using a very limited set of auxiliary inputs in that case?
So I was reading about Talotti, Paier, and Miculan - ECC’s Achilles’ Heel: Unveiling Weak Keys in Standardized Curves. The underlying idea is to lift the discrete logarithm problem to $\mathrm{prime}−...
- 87
2
votes
0
answers
123
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Polynomial discriminant equation
This is a fairly straightforward question, and I am hoping a definitive answer exists.
Does there exist a quadratic form $A \in \mathbb{C}[x_1, x_2, x_3, x_4]$ and a cubic form $B \in \mathbb{C}[x_1, ...
- 27k
7
votes
1
answer
569
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Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?
Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+...
CommunityBot
- 1
2
votes
1
answer
125
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Questions about elliptic curves with level-$n$ structure
Let $n$ be a positive integer, which is $4$ or a prime number $l$.
Let $E$ be an elliptic curve defined over a number field $K$.
Assume that all the $n$-torsion points of $E$ are defined over $K$, i.e....
- 8,945
1
vote
2
answers
197
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What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 103
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0
answers
141
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State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
4
votes
1
answer
183
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About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local fields
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be an elliptic curve defined over $K$. Tate's algorithm can be used to compute the Kodaira symbol of the reduction type of $E$.
However, I ...
- 37k
2
votes
1
answer
308
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Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?
Let $E/K$ be a number field. For quadratic field extensions $L/K$, it is known that $\operatorname{Ш}(E/L)[2]$ can be arbitrarily large (cf. P. L. Clark and S. Sharif, "Period, index and ...
- 149k
5
votes
1
answer
379
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Factorisation of division polynomial
Let $\Psi_n$ denote the $n$-th division polynomial associated with the elliptic curve $y^2 = x^3 + A$, where $n$ is a natural number. The division polynomials are defined recursively, as described in ...
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5
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2
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378
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Are there integer solutions of $m^4+m^2n^2+n^4=k^2$?
The recent question about Sets of integers with same sum and same sum of reciprocals, with its wealth of solutions, raises naturally the question: Can we require all those numbers to be squares? Of ...
- 22.6k
6
votes
0
answers
135
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Reconstructing a scheme from its quotient stack
Let $X/S$ be a scheme over a base $S$, with a group action by a nice group $G/S$ (typically $G/S$ affine smooth).
Can we reconstruct $X$ from its quotient stack $[X/G]$?
It seems that we can expect $X$...
- 16.2k
1
vote
0
answers
137
views
Two elliptic curves with the same j-invariants
This is an interesting observation of mine when exploring moduli of elliptic curves.
Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
- 12.9k
0
votes
0
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190
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About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 4,713
7
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0
answers
141
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Average number of $\mathbb{F}_p$-points over twists of a variety
Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have:
Fact ...
4
votes
0
answers
130
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mod $p$ local Galois representation attached to elliptic curves
In the paper, lemma 4.4. The author gives the form of the representation of $G_p$ on $E[p]$ of the form
$$\begin{pmatrix} \varepsilon\chi & *\\0 & \chi^{-1} \end{pmatrix}.$$
Do they assumed ...
- 275
34
votes
7
answers
3k
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What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?
Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$.
The best empirical evidence for this hunch ...
- 4,745
2
votes
1
answer
138
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small monodromy for polarized Abelian families over a torus?
In Lawrence-Venkatesh, they proved in Prop 5.3 a
beautiful finiteness property for a locus, successfully avoiding the using of Tate conjecture. Note that the introduction of $size_v$ and friendly ...
- 149k
2
votes
0
answers
126
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Full level structure Deligne-Rapoport v.s. Katz-Mazur
For modular curves over schemes there are two main references that I use, namely Deligne Rapoport [DR], and Katz-Mazur [KM]. However I recently noticed that there is a difference in conventions in ...
- 2,224
1
vote
0
answers
157
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Degeneracy of the Cassels-Tate pairing $\operatorname{Ш}(E/K)[n]\times \operatorname{Ш}(E/K)[n]\to \Bbb{Q}/\Bbb{Z}$
$\DeclareMathOperator{\Sha}{Ш}$ Let $E/K$ be an elliptic curve over a number field $K$. Let $\Sha (E/K)$ be the Tate-Shafarevich group, and let $n\ge 2$ be an integer. According to Theorem 15 in the ...
- 35.5k
5
votes
2
answers
319
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Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$
First I shall begin by laying out some notation (I shall be using the conventions that are used by both DLMF and Mathematica which occasionally differ from the standard literature):
Let $\Lambda:=\...
- 533
7
votes
1
answer
626
views
Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$
I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, ...
- 2,858
6
votes
1
answer
407
views
Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 10.7k
2
votes
1
answer
240
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Cohomology of torsion points on elliptic curves
$\DeclareMathOperator\Gal{Gal}$Let $E$ be an elliptic curve defined over a number field $K$. Put $G:=\Gal(\bar{K}/K)$, and for each valuation $v$ of $K$, put $G_v:=\Gal(\bar{K_v}/K_v)$. Consider the ...
- 12.9k
2
votes
0
answers
43
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The vector space dimension of Selmer group of abelian variety
Let $A/K$ be an abelian variety with $\mathbb{Z}[\mu_p] \subset End_K(A).$
Let $\pi$ be the prime of $p$, i.e. $(p)=(\pi^{p-1})$.
I want to obtain the relation of Selmer groups $Sel_\pi(A/K)$ and $...
- 85
2
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0
answers
78
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Detect all isogenies of an elliptic curve over a given number field
Given $K$ a number field and $E/K$ an elliptic curve, is there an algorithm which gives all the elliptic curves $F/K$ isogenous to $E$ (up to isomorphism)?
Or is there a bound on how many $F/K$ are ...
- 7,127
1
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0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
7
votes
1
answer
193
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Constructing non-split $\mathbf{G}_m$-extensions of elliptic curves
I would like to find examples of abelian surfaces $A$ over a DVR $R$ with residue field $k$, so that the special fibre of $A$ is a non-split $\mathbf{G}_{m/k}$-extension of an elliptic curve over $k$.
...
- 149k
0
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1
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136
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Multiplication-by-m map as an isogeny
I'm reading Luca De Feo's slides; on the slide 12 it's written that the multiplication map $[m] : E \rightarrow E$ is an isogeny. I do not understand why this isogeny is non-cyclic. At the first ...
1
vote
1
answer
131
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Reduction of elliptic curves over local fields
Let $E$ be an elliptic curve defined over a p-adic local field $K$, with $j$-invarient $j(E)\in K$. Let $\mathscr{O}_K$ be the ring of integer of $K$. If $j(E)$ does not belong to $\mathscr{O}_K$, ...
- 4,745
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1
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243
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On the degeneration of the elliptic surface $E(n)$
The following matter should be widely known (if true). I am sorry for my ignorance!
For the natural $n$, let $E(n)$ be the corresponding elliptic surface.
In the analytic world, there exists a well-...
- 153
1
vote
1
answer
120
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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
Let $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$.
I'm searching for a reference for the statement of the ...
- 8,945
3
votes
1
answer
114
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The cyclic twist of elliptic curve is a principally polarized abelian variety
Let $L/K$ is a cyclic extension of degree $p$, and let $E/K$ be an elliptic curve.
Let $E^L$ be the kernel of the map $Res^L_{K}(E) \rightarrow E$, where $Res^L_{K}(E)$ is the Weil-restriction.
Is the ...
- 13k
9
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1
answer
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Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
- 33.8k
10
votes
2
answers
404
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Impact of the squarefreeness of the level for modular forms
I often notice papers and results that assume that the level is squarefree in the setting of modular forms, but have a hard time figuring how where this impacts or simplifies the argument. Is there in ...
- 23k
16
votes
5
answers
8k
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Is the ABC conjecture known to imply the Riemann hypothesis?
I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
- 13
6
votes
3
answers
2k
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Direct proof of special case of Hasse's theorem for elliptic curves
Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$.
If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is ...
- 12.3k
1
vote
0
answers
69
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Is it in theory possible to perform general Miller’s algorithm inversion as used with the optimal ate pairing with large trace in subexponential time?
Let’s I have the following :
2 curves $G_1$ defined on $F_p$ and $G_2$ being the $G_1$ curve’s twist defined on $F_p^2$ both having the same prime order ; a large trace ; and $F_p^{12}$ as their ...
- 87
16
votes
2
answers
1k
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Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$
Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives:
$$ \#E(\mathbb F_p) = 1+ \sum_{x=0}^{p-1} \left(...
- 12.3k
3
votes
1
answer
180
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Galois cohomology for rational torsion of elliptic curves
Let $E/K$ be an elliptic curve over a number field.
Let $M=K(E[p])$.
I want to know $H^1(M/K,E[p])$: for $p=2$, it is $0$, but what about the case $p>2$?
Is it always zero?
In fact, I want to know ...
- 63.9k
3
votes
1
answer
414
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Twists of elliptic curves
I have a few questions regarding twists of elliptic curves.
In the context of the Shafarevich group, I see people refer to the group of twists of an elliptic curve $E/\mathbb{Q}$ by $H^1(\mathbb{Q}, ...
- 149k
2
votes
1
answer
147
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Finiteness and bounds for elliptic curves realizing a given galois representation
Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $...
- 37k
25
votes
3
answers
3k
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Background for the Elkies-Klagsbrun curve of rank 29
Elkies and Klagsbrun have recently announced an elliptic curve over Q of rank (at least) 29, improving on the previous record from 2006!
https://web.math.pmf.unizg.hr/~duje/tors/z1.html
It has trivial ...
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