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Results tagged with ag.algebraic-geometry
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user 70751
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
2
votes
1
answer
160
views
Local nontriviality of genus-one curves over extensions of degree dividing $6^n$
Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}_{p}$ whose degree divides a power of $6$, so that
$C(K)$ is not empty?
(I …
- 1,727
3
votes
1
answer
347
views
A question on the cohomology of elliptic curves over local fields
Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ove …
- 1,727
1
vote
0
answers
81
views
Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$
I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when:
(1.) $q=p$
and/or
(2.) $E$ has multiplicative reduction at $q$.
Here, $E$ is an ellip …
- 1,727
3
votes
0
answers
167
views
A question regarding Corollary 4.12 in Mumford's "Analytic construction of deg. Ab. Var."
Let $S=Spec A$ be the spectrum of an integrally closed, excellent and Noetherian ring. In his paper, David Mumford constructs an $S$-group scheme $G$. He shows that the torsion of $G_s$ is $p$-divisib …
- 1,727
-3
votes
1
answer
232
views
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$? [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}.$
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$?
- 1,727
0
votes
0
answers
235
views
Reference request: Détailed explanation why the Grassmannian scheme represents the Grassmann...
Similar questions have been asked on this site, including by myself, but none of these have been given a satisfying answer.
The question is: Why does the Grassmannian scheme represent the Grassmannian …
- 1,727
3
votes
0
answers
74
views
Beilinson-Lichtenbaum conjecture for algebraic extensions of $\mathbb{Z}/m$
Let $X$ be smooth over some field $k$ and $m\in\mathbb{Z}$ so that $m$ maps to a unit in $k^{\times}$. By Beilinson-Lichtenbaum one has an isomorphism of cohomology groups
\begin{equation*}
\opera …
- 1,727
0
votes
0
answers
376
views
Conditions for splitting of short exact sequence?
Assume $K$ is a number field and $E$ is an elliptic curve defined over $K$.
Are there conditions under which the short exact sequence
$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarro …
- 1,727
2
votes
0
answers
237
views
Reference request: Cohomology of Elliptic Curves
Is it true that the group
$$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$
is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility?
He …
- 1,727
1
vote
0
answers
82
views
Spectral sequences associated to cohomologies of simplicial type and derived-functor type: P...
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $ …
- 1,727
2
votes
0
answers
146
views
$L$-series and Riemann zeta function
I am currently reading SGA 4$\frac{1}{2}$, exposé 2: Rapport sur la formule des traces.
The $L$-series associated to a scheme $X$ of finite type over $\mathbb{F}_{p}$ is defined as
$$L(X,s):=\prod_{x\ …
- 1,727
2
votes
0
answers
82
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.) $\operato …
- 1,727
5
votes
0
answers
303
views
Existence of infinitely many Heegner points that are divisible by $p^{n}$ in $K_{\lambda}$
Let $E$ be an elliptic curve over $\mathbb{Q}$, $p>2$ a prime and $n,s\in\mathbb{N}$.
For $j\in\{1,...,s\}$ let $n_{j}\in\mathbb{N}$ be a natural number which may or may not be coprime to $p$.
Let $K/ …
- 1,727
0
votes
0
answers
90
views
Are the zeroes of the finite characteristic zeta functions dense in $\left\{s\in\mathbb{C}\m...
If $p$ is a prime, $n\in\mathbb{N}$ is a natural number and $C$ is a nonsingular curve over $\mathbb{F}_{p^{n}}$, the $\zeta$ function associated to $C\mid_{\mathbb{F}_{p^{n}}}$ is defined as
\begin{e …
- 1,727
2
votes
2
answers
570
views
Good reference on the algebraic geometry of non-associative rings
I am looking for a good reference about the algebraic geometry of non-associative rings. I am in particular interested in derivation algebras.
Preferrably an online resource or a book that is availabl …
- 1,727
6
votes
2
answers
362
views
Does the category of local rings with residue field $F$ have an initial object?
Let $F$ be a field. Does the category $C_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object?
This is, for instance, true if $F=\mathbb{F}_{p}$ for s …
- 1,727
0
votes
0
answers
125
views
Is $p^{-s}$ transcendental if $\zeta(s)=0$?
Let $K$ be a number field and $\mathcal{O}_{K}$ its ring of integers.
Let
$$\zeta_{K}(s)=\prod_{\mathfrak{m}}\frac{1}{1-\#(\mathcal{O}_{K}/\mathfrak{m})^{-s}}$$
be the $\zeta$ function associated to $ …
- 1,727
1
vote
0
answers
134
views
Grothendieck trace formula for schemes with étale fundamental groups that have no dense cycl...
This question may be more of a philosophical rather than mathematical nature.
Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\ma …
- 1,727
3
votes
0
answers
185
views
Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theo...
A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic …
- 1,727
1
vote
0
answers
60
views
Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion...
I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic.
I have come up with the following defini …
- 1,727
3
votes
1
answer
394
views
Selmer and free rank of Elliptic Curves
If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many implications of the Birch and Swinnerton-Dyer conj …
- 1,727
6
votes
1
answer
771
views
Relationship between Tate-Shafarevich group and the BSD conjecture
The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}_{q},$ this result is due to Kato and Trihan if I am not mistaken …
- 1,727
3
votes
1
answer
279
views
Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism
If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\Ome …
- 1,727
1
vote
1
answer
224
views
Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvabl...
Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over $\ov …
- 1,727
5
votes
2
answers
626
views
Reference request: Kleiman's proof of Snapper's Lemma
On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as
a special case of Snapper's Lemma, see "A …
- 1,727
2
votes
0
answers
118
views
Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta functi...
Let
\begin{equation*}
\zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}
\end{equation*}
be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation
\begin{equati …
- 1,727
3
votes
1
answer
233
views
Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-s …
- 1,727