All Questions
2,494 questions
5
votes
1
answer
362
views
Conductor of determinant of a 2-dimensional Galois representation divides conductor of representation
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)...
- 816
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. [...
- 1,446
0
votes
1
answer
272
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 ...
- 9,497
2
votes
2
answers
636
views
L-functions of Calabi-Yau varieties
This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ...
- 63.9k
3
votes
0
answers
215
views
Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
- 85
3
votes
0
answers
171
views
Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants
$\newcommand{\Sha}{\operatorname{Sha}}$Let $E/\mathbb{Q}$ be an elliptic curve, and let $\Sha(E/\mathbb{Q})$ denote the Tate–Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary ...
- 1,541
2
votes
1
answer
220
views
Looking for an example of a point $P$ on an abelian variety $X$ such that no curve on $X$ contains all multiples of $P$
Is there an example of an abelian variety $X$ defined over a number field $K$, with $\dim X > 1$, and a $K$-rational point $P$ on $X$, such that no curve $C$ on $X$ (say defined over a number field)...
- 149k
2
votes
1
answer
277
views
Understanding an example of abelian-type Shimura varieties
I'd like some help understanding the idea of abelian-type Shimura varieties. In paricular, I understand an abelian-type Shimura datum $(G,X)$ generally parameterizes non-rational Hodge structures ...
- 2,054
1
vote
0
answers
145
views
Multiplicity and the perfect projective line
Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.
Let $\Gamma$ be the ...
- 85
1
vote
1
answer
370
views
Self-intersection of the diagonal on a surface
Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
- 28.8k
5
votes
0
answers
261
views
Equations for conic del Pezzo surfaces of degree one
Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
- 241
2
votes
1
answer
223
views
Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring
Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
- 16.2k
9
votes
1
answer
426
views
Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/\Bbb{Q}$ is bounded by some constant $M$, for all integers $D \in \Bbb{Z}$?
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$.
Let $E_D$ be a quadratic twist of $E$ also defined over $\Bbb{Q}$.
Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/...
- 27k
0
votes
1
answer
152
views
Almost Pell type equation
Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
CommunityBot
- 1
0
votes
0
answers
129
views
Extend line bundle on regular curve to it's regular model
Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$.
Assume, $C$ admits a proper regular flat model $\...
- 5,986
1
vote
1
answer
218
views
Dimension of Zariski closure of a closed point of generic fiber
Let $S= \operatorname{Spec} A$ be a local Dedekind scheme of dimension $1$, (eg spectrum of localization at a prime of the ring of integers of a number field). Let $s \in S$ it's unique closed point ...
- 21.4k
3
votes
1
answer
469
views
Adic generic fiber of a small formal scheme in the sense of Faltings
$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
- 16.2k
2
votes
0
answers
134
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
6
votes
1
answer
370
views
Interpreting group-theoretic sentences as statements about algebraic groups
Suppose we are given a sentence in the language of groups, e.g. $\phi=\forall x\forall y(x\cdot y=y\cdot x)$, and suppose that we are also given the data defining an algebraic group $G/k$. One can ...
- 333
2
votes
0
answers
354
views
Square-zero extensions mod $p^n$
$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
7
votes
0
answers
174
views
Failure of injectiveness of maps between cotangent spaces of abelian varieties
Let $p$ be a prime and $K$ a finite extension of $\mathbb Q_p$ with ramification index $e$. Let $\mathcal O_K$ be the ring of integers of $K$ and $k$ its residue field and the unique maximal ideal. ...
- 2,224
2
votes
0
answers
275
views
Is there any relation between Berkovich spaces over $\Bbb Z$ and Arakelov theory?
As I understand it, both Arakelov geometry and Berkovich geometry over $\Bbb Z$ (or $\mathcal O_K$) consider geometric objects that contain in some sense information about both archimdean and ...
- 804
2
votes
4
answers
618
views
A question on function fields (extending my previous question)
Consider the extension $\mathbb Q(a,b)$ of the field of rationals, where $a$, $b$ are algebraically independent transcendentals. To $\mathbb Q(a,b)$ adjoin the roots of the polynomials $x^5+a^5=1$ and ...
- 12.9k
1
vote
1
answer
260
views
On the estimate for a double exponential sum
I encounter a hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum:
$$...
- 149k
3
votes
1
answer
321
views
Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $
Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.
Consider the natural map
$$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{...
17
votes
3
answers
2k
views
Are some congruence subgroups better than others?
When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins ...
- 831
1
vote
1
answer
115
views
About the power of numbers primes distribution
Let $r>0$, $p\neq q$ two primes numbers and $A=\{(m,n)\in\mathbb N^2; |p^m-q^n|\leq r\}$.
Is it true that $A$ is a finite set?
- 3,319
1
vote
0
answers
79
views
A general theory of pairings
Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra.
There are also text books for bilinear forms and related quadratic ...
- 231
11
votes
0
answers
407
views
Algebraizability of formal schemes
Let $R$ be a complete DVR with quotient field $K$ and let $f:\mathfrak{X} \to \mathrm{Spf}(R)$ be a smooth proper formal scheme.
If the (rigid analytic) generic fibre of $f$ is (the analytification ...
- 9,497
1
vote
0
answers
162
views
Motivic complex on arithmetic schemes
If we believe the finite generation of motivic cohomology for regular arithmetic schemes like $X$ then we can see that (using Quillen-Lichtenbaum)for infinitely many primes $l$ we have an isomorphism ...
- 5,901
1
vote
0
answers
120
views
Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$
Let $p$ be a prime number and $n$ be positive integer.
Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve.
LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$.
This is the biggest size of $Sha(...
- 1,541
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 52.7k
0
votes
1
answer
160
views
Another generalisation of euclidean division on integers
Let $n \in\mathbb N^*$.
What are all the surjective functions $f: \mathbb N \rightarrow \{0,...,n-1\}=E $ such that there exist functions $g,h$ from $E^2$ to $E$ with:
$\forall (m,k) \in\mathbb N^2,f(...
- 109k
46
votes
2
answers
4k
views
What are the potential applications of perfectoid spaces to homotopy theory?
This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ...
- 1,838
3
votes
0
answers
227
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\...
- 371
7
votes
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 2,368
3
votes
0
answers
132
views
How does the number of connected components of the Néron model change in a family of abelian varieties?
Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
- 333
4
votes
1
answer
736
views
Picard group of quasi-projective varieties
Let $X$ be a smooth open sub-variety of a projective, not necessarily smooth, variety $X'$, defined over a finite field.
Is $\text{Pic}(X)$ a finitely generated abelian group?
I'm tempted to just ...
- 461
3
votes
1
answer
393
views
On explicit examples of the Parshin Construction
In the Proof of Mordell Conjecture by Gerd Faltings, it is famous that Parshin constructed a curve $C_P$ for each $\mathbb{Q}$-rational point $P$ on the given curve $C$ over $\mathbb{Q}$ such that the ...
CommunityBot
- 1
2
votes
0
answers
259
views
The group of the modular automorphisms of the Shimura curves
Let $B$ be a rational indefinite division quaternion algebra, $(X,G)$ the Shimura datum associated with $B$ (i.e., $X$ is the upper half plane and $G(R) = (B \otimes_\mathbb{Q} R)^*$ for a ring $R/\...
- 1,364
1
vote
1
answer
238
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 ...
- 1,386
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 20.5k
8
votes
2
answers
424
views
$n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$
Let $E/\mathbb{Q} = E_{a,b}$
$$\displaystyle y^2 = x^3 + ax + b, a,b \in \mathbb{Z}$$
be an elliptic curve defined over the field of rational numbers, and let $n \geq 3$ be an integer. Let $K_n$ be ...
- 10.9k
4
votes
2
answers
448
views
$p$-divisibility of Picard groups
Let $p$ be a prime number and let $k$ be a field with $char(k)\neq p$ such that all finite extensions have degree coprime to $p$. (For example, we can take $k=\mathbb{R}$ and $p\neq 2$ or let $k$ the ...
- 10.5k
5
votes
1
answer
389
views
Fermat cubic hypersurfaces over finite fields
Consider the Fermat cubic
$$
X = \{x_0^3+\dots +x_n^3 = 0\}\subset\mathbb{P}^n_{\mathbb{F}_{q}}
$$
over a finite field $\mathbb{F}_{q}$ with $q$ elements.
If $q \equiv 2 \mod 3$ then the projection $\...
- 14.7k
4
votes
1
answer
240
views
Points on affine hypersurface over finite field
I am interested in the hypersurface $X\subset\mathbb{A}^4_{\mathbb{F}_{5^n}}$ defined by
$$
X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\}
$$
over a finite field $\mathbb{F}_{5^n}$ with $5^n$ elements. Via ...
- 5,958
4
votes
1
answer
183
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 ...
- 47.4k
4
votes
2
answers
865
views
Extending finite morphisms of curves to finite morphisms of arithmetic surfaces
Let $\pi:Y\longrightarrow X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Let $h:\mathcal{X}\longrightarrow \textrm{Spec} \ O_K$ be a regular ...
- 9,497
4
votes
0
answers
284
views
Intermediate arithmetic results in F_1 geometry
Much is made of the search for a proof of the Riemann hypothesis via the field with one element. Are there any lesser classic arithmetic results that have been proved with F_1 geometry, such as the ...
- 447
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 12.9k