All Questions
2,494 questions
1
vote
1
answer
445
views
What is the reduction of this hyperelliptic curve
Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose ...
- 1,213
4
votes
2
answers
402
views
Defining isogenies over smaller fields
I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts ...
- 1,213
7
votes
2
answers
494
views
Jacobians defined over smaller fields
Let $L/K$ be an extension of number fields.
Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$.
In general, the Jacobian $J(X)$ probably doesn'...
- 1,213
12
votes
2
answers
1k
views
Modularity of higher dimensional abelian varieties
In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, one can give a proof using ...
- 7,072
9
votes
5
answers
2k
views
The significance of modularity for all Galois representations
On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...
- 7,072
3
votes
0
answers
406
views
Is the geometry of a variety determined by the counts of rational points?
In Diophantine Geometry: An introduction, Hindry and Silverman write "Geometry Determines Arithmetic" (pg. 2) and "Geometry Governs Arithmetic" (pg. 474).
On pg. 211 of the same book, the authors ...
- 7,072
2
votes
1
answer
615
views
Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?
For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...
- 945
1
vote
0
answers
192
views
"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
32
votes
4
answers
5k
views
Over which fields does the Mordell-Weil theorem hold?
According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...
- 20.8k
1
vote
0
answers
187
views
Fields over which cubic hypersurfaces are rational
All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...
- 3,779
10
votes
1
answer
749
views
Is the Hasse principle a birational invariant?
Is the Hasse principle a birational invariant?
It is probably a very trivial question, but I am a beginner in arithmetics.
- 3,779
8
votes
1
answer
758
views
Honda-Tate in families
Let $k$ be a finite field, say with $q=p^a$ elements. Honda-Tate theory states that there is a bijection between isogeny classes of simple abelian varieties over $k$ and $\mathrm{Gal}(\overline{\...
- 5,670
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
4
votes
1
answer
559
views
Heuristic for the Fermat-Catalan conjecture
[Edit: I've since realized that my question is confused: in particular, the minimum value of k that you need to sum from increases with the largest exponent under consideration so that the sum over ...
- 7,072
7
votes
0
answers
353
views
Counting higher dimensional abelian varieties of a given conductor
This question is a follow up to an earlier question of mine on enumerating elliptic curves of a given conductor.
I've heard people say that studying higher dimensional varieties via explicit ...
- 7,072
3
votes
1
answer
287
views
$K$-groups and dual graphs of special fibers
Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let $\mathcal{C}...
- 337
9
votes
2
answers
3k
views
Isogeny classes of elliptic curves
Let $E \subset \mathbb{P}_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in \...
- 3,555
10
votes
2
answers
1k
views
Finiteness of elliptic curves of a given conductor
It follows from the modularity theorem for elliptic curves over $\mathbb{Q}$ that there are finitely many elliptic curves of a given conductor $N$. Moreover, one can algorithmically enumerate them. [...
- 7,072
14
votes
1
answer
1k
views
Frobenius splitting of Fano varieties
Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
- 16.2k
15
votes
6
answers
3k
views
Characteristic zero and characteristic $p$ in algebraic geometry
Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known ...
user16974
0
votes
1
answer
273
views
local galois representation with higher coefficient
Suppose K is a local field , G is its galois group, V a fine dimensional Vector space over F, which is a sub field of K, and totally ramified over $Q_p$. Consdider the linear action of G on V (V is ...
- 709
2
votes
0
answers
637
views
Algebraicity of power series over the rationals from the algebraicity over Fp
Van der Poorten conjectures [in "Power series representing algebraic functions," Sem. Th. Nombres Paris 1990-91] that if a power series over the rationals is the [complete] diagonal [of a rational ...
- 527
5
votes
1
answer
630
views
Special value of $L$-function
Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...
- 471
7
votes
1
answer
749
views
Examples of finiteness of rational points for hypersurfaces in $\mathbb P^3_{\mathbb Q}$ of degree $>4$
Given an homogeneous polynomial $F(X,Y,Z,T)\in \mathbb Q[X,Y,Z,T]$ of degree $>4$, the surface it defines is well-known to be of general type. Suppose, moreover, that this surface doesn't contain ...
- 1,386
5
votes
2
answers
356
views
Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places?
See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a ...
user19475
3
votes
0
answers
315
views
Question about witt vector of some ring
Suppose $R=Z_p[t]$ , and $\hat{R}$ its p-adic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And $R_{\infty}=...
- 709
4
votes
0
answers
282
views
Does semi-stable reduction behave well with Weil restriction of scalars
Let $A$ be an abelian variety over a number field $K$ with semi-stable reduction over $O_K$.
Does the Weil restriction $\textrm{Res}_{K/\mathbf{Q}}A$ of $A$ to $\mathbf{Q}$ have semi-stable reduction ...
- 1,213
11
votes
1
answer
761
views
Mordell-Weil group of the universal abelian scheme
Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
...
- 3,076
0
votes
2
answers
336
views
Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dimension smaller than $g$ at some point
Let me be more precise than the title. (This will be my last attempt to do something with abelian varieties. Sorry for all the basic questions. The answers have been great!)
Let $A$ be a simple ...
- 1,213
0
votes
1
answer
298
views
Is any simple abelian variety covered by a non-simple abelian variety
Let $A/k$ be a simple abelian variety.
Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$?
I don't need $f:B\to A$ to be etale.
- 1,213
3
votes
1
answer
492
views
Are abelian varieties degree two covers of some projective space
Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.
There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question.
Does there exist a finite morphism $A\to \mathbf{P}^...
- 1,213
20
votes
3
answers
2k
views
what is the maximum number of rational points of a curve of genus 2 over the rationals
Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are ...
- 209
1
vote
1
answer
215
views
Is the number of twists of a curve with a section in a given field finite
Let $X$ be a smooth projective geometrically connected curve over a number field $k$ of genus $g\geq 2$.
Is the number of twists of $X$ always infinite? (The answer is no, because there aren't any ...
- 281
27
votes
6
answers
4k
views
Does the moduli space of smooth curves of genus g contain an elliptic curve
Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.
Does $M_g$ contain an elliptic curve?
The answer ...
- 281
6
votes
1
answer
302
views
Is the class of $k$-gonal curves dominant
Before I start, let me make a note on terminology. Curves are always smooth projective connected curves over an algebraically closed field of characteristic zero.
Let $\mathcal C$ be a class of ...
- 9,497
3
votes
1
answer
485
views
Group of connected components of the global Néron-Raynaud model of a torus
Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$
defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus.
We choose one ...
- 359
3
votes
0
answers
145
views
Curves whose stable reductions do not contain rational curves
Let $X$ be a smooth projective curve over $K:=K(A)$. $A$ is a strict henselian ring, $A/m=k=\bar k$. Suppose $\cal X$ is a stable model of $X$, ${\cal X}_{s}$ is the special fiber.
My question is:
...
- 1,921
7
votes
2
answers
1k
views
questions on Néron-Tate canonical height
I have three questions regarding height pairings:
In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function:
"Let $V/R$ be a ...
user19475
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
4
votes
0
answers
413
views
On Stickelberger's Theorem over function fields
Here is the setup to Stickelberger's theorem over number fields (following Washington's book Intro. to cyclotomic fields).
Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. ...
- 293
0
votes
1
answer
472
views
surjectivity of rational points induced by surjective map from affine space
Let $k$ be a local field of char $0$ (which is the case I concern).
Let $V$ be a variety defined over $k$ and
let $f: \mathbb A^n\to V$ be a surjective map
(over the algebraic closure of $k$) ...
- 853
3
votes
2
answers
662
views
Moduli Space of Abelian Varieties with a N-torsion point
Does there exists (as scheme, or as some sort of stack) the moduli space of principally polarized Abelian Varieties together with a point of order $N$, for $N>1$ an integer?
In the case of ...
- 1,386
3
votes
1
answer
667
views
What is an automorphic representation of CM type ?
In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a ...
- 809
9
votes
2
answers
1k
views
On Grothendieck's period relations
Let $V$ be a smooth projective variety defined over $\mathbf{Q}$ and denote by
$$
\omega: H_{dR}^*(V,\mathbf{Q})){\otimes_{\mathbf{Q}}}\mathbf{C}\rightarrow H_{B}^*(V,\mathbf{Q})\otimes_{\mathbf{Q}}\...
- 7,609
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 3,605
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 3,362
13
votes
2
answers
572
views
Existence of points on varieties which avoid a given number field.
Let C be a geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that
$L \cap K' = K$, and
$C(L) \neq \...
- 10.5k
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
2
votes
0
answers
622
views
The cohomology of the relative dualizing sheaf of a relative curve
Let $X\to S$ be a curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$.
I know that $\...
- 1,213
1
vote
0
answers
187
views
stack quotient question
Hi,
I have the following question:
let $k$ a field with $char(k)= p>0$, which we can assume to be perfect, $W(k)$ the ring of Witt vector, and $a,b$ positive integers.
Consider the ring $R=W(k)[x,...
- 11