All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
0
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179
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Points at which a polynomial becomes reducible
Let $n \geq 10$ and set $\mathbf{y} = (y_1,\ldots,y_n)$. Let $Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y})$ be non-zero quadratic forms with integer coefficients such that the cubic form $x_1Q_1(\mathbf{y})+...
- 105
2
votes
1
answer
437
views
Sheaf cohomology in number theory
I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
- 186
1
vote
1
answer
315
views
About simple motives
I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...
user497064
6
votes
0
answers
629
views
On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"
I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
- 507
2
votes
0
answers
197
views
Mumford's computation of the determinant of cohomology of a relative curve
In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR ...
- 2,044
12
votes
2
answers
2k
views
What is the Perrin-Riou logarithm (or regulator)?
Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this workshop at the CRM in ...
- 3,309
0
votes
0
answers
182
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Does a $p$-adic power series $F(x,y)=\sum_{i,j \geq 0}b_{ij}x^iy^j \in \mathbb Z_p[[x,y]]$ have finitely many zeros in $\mathfrak{m}_{\mathbb C_p}$?
Let us consider the $p$-adic field $\mathbb Q_p$ with ring of integers $\mathbb Z_p$ and maximal ideal $\mathfrak{m}$.
Then any $p$-adic power series $f(x)=\sum_{n>0}a_nx^n \in \mathbb Z_p[[x]]$ ...
- 930
7
votes
0
answers
296
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Connections between Borger's absolute geometry and Connes' and Consani's $\Gamma$-spaces
As the idea of an absolute geometry over the field with one element $\mathbb{F}_1$ becomes more clear, two approaches seem to have crystallized, being based on different assumptions and going into ...
1
vote
0
answers
191
views
Vanishing of the local étale cohomology sheaf (?)
Let $X$ be a locally noetherian regular scheme, and let $Z$ be a closed subscheme of $X$ whose codimension $d > 0$ at every point.
Let $U$ be the complement of $Z$ in $X$.
For a sheaf $\mathscr{F}$ ...
- 185
6
votes
0
answers
221
views
Motives in tropical geometry
Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.
- 163
2
votes
1
answer
131
views
Adjacent reducible polynomials
Let $P[X_1, X_2, \ldots, X_N]$ be a reducible polynomial in $\mathbb{Z}[X_1, X_2, \ldots, X_N]$ such that $P[X_1, X_2, \ldots, X_N] + 1$ is also reducible. What (if anything) can we say about $P$?
One ...
- 1,703
7
votes
1
answer
315
views
Rational points on regular curves over global fields
Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...
- 21.3k
64
votes
6
answers
52k
views
Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture
Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and ...
- 2,624
2
votes
0
answers
142
views
Upper bound for the torsion subgroup of an elliptic curve over arbitrary number fields
Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn ...
- 565
1
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0
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137
views
What is the lattice of the field $\mathbb Q_p(\sqrt[p^5-1]{p^2})$?
Let $p$ be odd prime and $\mathbb Q_p$ be the $p$-adic field. Consider the field extension $K=\mathbb Q_p(\sqrt[(p^5-1)]{p^2})$ of $\mathbb Q_p$ of degree $\frac{p^5-1}{2}$.
My question:
I want see ...
- 930
1
vote
0
answers
95
views
An elliptic threefold and the Mordell–Weil lattices of its reductions
Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\...
- 1,833
27
votes
4
answers
3k
views
Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
- 1,969
5
votes
0
answers
192
views
On the elementary proof of Dirichlet theorem on arithmetic progressions
In [Cassels, JWS, Rational quadratic forms, p. 333], the autor says: "In fact the elementary proof of Dirichlet's theorem [Selberg (1949)] makes essential use of the existence of genera".
In ...
17
votes
1
answer
2k
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How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?
In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
- 3,309
2
votes
0
answers
146
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Can we say anything about the zeros and Galois group of the polynomial $(x^p-a)^{p^2}-p^{p^2+1}x+p^{p^2} a=0$?
Let $p$ be an odd prime number and $\mathbb Q_p$ be the $p$-adic number field. Let $K=\mathbb Q_p(a)$ be the extension by $a=p^{\frac{p^2+1}{p^3-1}}$.
Consider the polynomial $f(x)=(x^p-a)^{p^2}-p^{p^...
- 930
3
votes
1
answer
249
views
Completely reducible subgroups over local field in terms of closed orbits
$\DeclareMathOperator\GL{GL}$Let $ \overline{\mathbb{Q}}_{p} $ be an algebraic closure of $ p $-adic numbers $ \mathbb{Q}_{p} $. A closed subgroup $ H $ of a general linear group $ \GL_{n}(\overline{\...
- 667
4
votes
0
answers
252
views
Lifting the connected-etale sequence of the $p$-torsion of an elliptic curve
Suppose that $R$ is a complete DVR with characteristic 0 fraction field $K$, maximal ideal generated by $p$ and characteristic $p>0$ residue field $k$ which is algebraically closed. Suppose that $\...
- 305
6
votes
1
answer
445
views
Why do Chern forms show up in Arakelov geometry?
Let $X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on $X$. In order to define the height of an integral closed subset $Y$ we define it on closed ...
- 193
5
votes
0
answers
237
views
Hyperelliptic curve with prescribed rational points?
Given a set of rational points $S$, does there always exist a hyperelliptic curve $C$ such that $C(\mathbb{Q})=S$?
Namely, which sets could arise as the set of rational points of a hyperelliptic curve?...
- 203
4
votes
0
answers
211
views
Rational solutions to Catalan's equation
Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation
$$
x^{a}-y^{b}=1.
$$
for $a, b > 1$ and $x, y > 0$ is $x = 3,...
- 7,182
4
votes
1
answer
442
views
Trigonometric Diophantine equation
Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number?
This ...
- 1,462
1
vote
0
answers
144
views
Elliptic curves whose $2,3,5$-parts of Sha are large
Let $E$ be an elliptic curve, and $\text{Sha}(E)$ its Shafarevich-Tate group which measures the failure of the local-to-global principle for the curve. It is conjectured that $\text{Sha}(E)$ is a ...
- 26.9k
62
votes
9
answers
9k
views
There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?
Quadratic forms play a huge role in math. This leads one to wonder: Is there a theory of cubic forms, quartic forms, quintic forms and so on? I have failed to discover any. Is there any such theory? ...
- 705
2
votes
0
answers
148
views
Can there exist different smooth, proper schemes over the p-adics with the same generic fiber? [duplicate]
Can there exist smooth, proper $X_1,X_2/\mathbb Z_p$ such that their generic fibers are isomorphic but their reductions mod $p$ are not? Are there examples if we insist that the special fibers are ...
- 7,746
3
votes
1
answer
470
views
Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0?
The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian ...
- 193
3
votes
1
answer
296
views
$p$-power torsion of semiabelian variety
Let $K$ be a finite extension field of $\mathbb{Q}_p$. Let us consider a semiabelian variety $G$ defined over $K$, i.e there exists an extension of an abelian variety $B$ and a torus $T$ defined over $...
- 247
3
votes
0
answers
210
views
How to find rational points on genus 2 rank 2 curves such as $y^2=x^6-4x+4$?
The question is in the title. The motivation comes from trying solving Diophantine equations in order, see Can you solve the listed smallest open Diophantine equations? . Because there is an algorithm ...
- 7,182
0
votes
0
answers
125
views
Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?
Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
- 617
1
vote
0
answers
96
views
Locally symmetric spaces dependence on number field
A special case of locally symmetric spaces is the moduli space of abelian varieties of a given dimension $g$ (over a given base field $k$), lets call it $\mathcal{A}_g$ and is a $k$-scheme. For any ...
- 4,408
5
votes
1
answer
430
views
Where can I find that Weil suggested a cohomology theory for characteristic $p>0$?
I have seen that in Grothendieck's paper "THE COHOMOLOGY THEORY OF ALGEBRAIC VARIETIES", he says "The need of a theory of cohomology for 'abstract' algebraic varieties was first ...
- 121
1
vote
2
answers
349
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
1
vote
0
answers
113
views
Counting Points on a Plane Curve
I want to find the number of points of $\displaystyle \mathcal{C}_{h} \cap A^{2}( F_{q})$, (char(F) is not equal to 2,3), where $\displaystyle h$ is a rational function defined as $\displaystyle h( x) ...
- 11
2
votes
1
answer
571
views
What goes wrong with this proof of Dirichlet's Theorem?
I am curious what goes wrong with this invariant theoretic argument of Dirichlet's Theorem of primes in arithmetic progression. I know that something goes wrong, but I am really curious about what ...
- 912
0
votes
0
answers
125
views
Is there a kind of uniqueness of Poincaré duality? [duplicate]
There is a related question. I would like to know if there is a more intrinsical way to show this. I want to know if we can get this through the uniqueness of Poincaré duality or the comparison ...
- 121
7
votes
1
answer
916
views
Why are Shimura varieties the "right" objects?
So this is probably blasphemist to ask and I've resisted asking this for a while. Essentially my question is why are locally symmetric spaces/Shimura varieties the "right" object to study ...
- 4,408
10
votes
1
answer
3k
views
Roadmap to understand the Scholze's proof of the local Langlands correspondence for $\text{GL}_n$ over $p$-adic fields
I would like to know which books I should read to understand the paper "The local Langlands correspondence for $\mathrm{GL}_n$ over $p$-adic fields" written by Peter Scholze.
I only know ...
- 111
40
votes
1
answer
10k
views
What actually is the idea behind the condensed mathematics?
Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead ...
- 1,525
6
votes
1
answer
512
views
Mordell conjecture over function fields
So I've read (for instance in the introduction to R.S de Jong's thesis ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection ...
- 4,408
3
votes
1
answer
384
views
Overconvergent modular forms and the level at $p$
I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.
The ...
- 241
10
votes
0
answers
2k
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Roadmap for p-adic Hodge theory
I'd like to be able to start studying p-adic Hodge theory and hope that by posing this question, I can be better prepared to work towards it. I ask for a roadmap because I understand that I have a lot ...
- 544
6
votes
0
answers
590
views
Affine GIT quotients and the excursion algebra in Fargues–Scholze
Some background:
Let us fix a non-archimedean local field $E$ with residue characteristic $p$, and let $G$ be some connected reductive group over $E$. In [FS, §VIII.1.1] the authors define a moduli ...
- 777
4
votes
1
answer
354
views
Can a general quintic be solved using inverse beta regularized function?
Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
- 10.1k
8
votes
1
answer
2k
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Some questions from the paper by Scholze-Weinstein
The following is from the paper by Scholze-Weinstein on moduli of $p$ divisible groups.
My question is from a part of Lemma 4.1.7: If $R$ is a semiperfect ring, then the canonical map $W(R^{\flat}) \...
- 81
3
votes
0
answers
175
views
Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology
Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...
- 863
2
votes
0
answers
134
views
Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?
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{...
- 1,805