Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,042
questions
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Is the Dieudonne module actually a cohomology group?
One often times thinks of the Dieudonne module $M(X)$ of a $p$-divisible group (say over $k$, a perfect characteristic $p$ field) as being some sort of cohomology theory
$$M:\left\{p\text{- divisible ...
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7
answers
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SAT and Arithmetic Geometry
This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...
18
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1
answer
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Examples of solid abelian groups
I am reading through Clausen's and Scholze's Lectures on condensed mathematics. I am struggling to understand the concept of solid abelian groups so I am looking for some examples.
Is the underlying ...
18
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1
answer
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Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?
By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!
18
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1
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Does the moduli space of genus three curves contain a complete genus two curve
Inspired by the question
Does the moduli space of smooth curves of genus g contain an elliptic curve
and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth ...
18
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1
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The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
18
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1
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On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result?
Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A_m$ denote the group of elements $a \in A(\...
18
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2
answers
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Galois representations attached to newforms
Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposition group at $p$. ...
18
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1
answer
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Arithmetic motivations for modularity in higher rank
The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching ...
18
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3
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A-valued points of projective space
I have been reading "The Geometry of Schemes" by Eisenbud and Harris and have a question about Exercise III-43. There, one should show that there is a bijection between the sets
$\{(n+1)\mbox{-tuples ...
18
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2
answers
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Divisibility properties of Hurwitz numbers
Define numbers $H_k$ for integers $k\geq 4$ by $\sum_{x \in \mathbf{Z}[i]}x^{-k}=\frac{H_k}{k!} \omega^k$, where $\omega=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}$. These are nonzero when $4|k$, and Hurwitz ...
18
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Eisenstein series as sections of line bundles on moduli spaces
It is well known that a modular form of weight k and level \Gamma is a global section of k-power of a Hodge line bundle over some modular curve. e.g. H^0(X,E^k).
My question is
How to characterize ...
18
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1
answer
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To what extent are modular parametrizations expected to generalize?
By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...
18
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1
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Discrepancy in Magma's calculation and Sage's of elliptic curve?
$\DeclareMathOperator\Sha{Sha}$I calculated the Tate–Shafarevich group $\Sha(E/K)[2]$ of the elliptic curve $E:y^2=x^3+17x$ over $K=\Bbb{Q}(\sqrt{-37})$.
I calculated that by hand and I reached the ...
18
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3
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Do isogenies with rational kernels tend to be surjective?
Dear MO Community,
this is a pretty vague title, so let me tell you the precise observation I have made.
Consider the family of elliptic curves over $\mathbf{Q}$ having a rational $5$-torsion point ...
18
votes
1
answer
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Field with one element example?
$$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$
This formula holds for the unit ball in $\mathbb{Q_p}$. This formula also holds for
$\mathbb{R}$ when $p=1$. Should one expect $$\...
18
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1
answer
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Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?
Let $X$ be a variety over a $p$-adic field $K$.
Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...
18
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0
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Cycles in algebraic de Rham cohomology
Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
17
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2
answers
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Why is one interested in the mod p reduction of modular curves and Shimura varieties?
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
17
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3
answers
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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 ...
17
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2
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Deligne Weil II
Deligne's Weil I has been published under the title "La conjecture de Weil: I" in 1974, and Weil II in 1980. So did Deligne know in 1974 that there would be a Weil II, and can one explain the period ...
17
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1
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Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?
From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...
17
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3
answers
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PNT for general zeta functions, Applications of.
When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results.
We talk of ...
17
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1
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Why is the section conjecture important?
As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
17
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2
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Geometric interpretation of Hida isomorphism
[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely.
As the title says, I would like to understand ...
17
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4
answers
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Usage of étale cohomology in algebraic geometry
I'm a student interested in arithmetic geometry, and this implies I use étale cohomology a lot. Regarding its definition, étale cohomology is a purely algebro-geometric object. However, almost every ...
17
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1
answer
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Construction of abelian varieties from Hilbert modular forms?
Some experts tell me that the construction of abelian varieties from
Hilbert modular forms is an (apparently difficult) open problem. However,
in view of the construction of $l$-adic Galois ...
17
votes
1
answer
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Conjectures of Peter Scholze about q-de Rham complex: examples
Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper
Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–...
17
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1
answer
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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 ...
17
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1
answer
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Why should the number of $\mathbb{F}_q$ points on degree $d$ curves $C\subset \mathbb{P}_{\mathbb{F}_q}^n$ decrease as $n$ increases?
This question concerns some counterintuitive results (to me at least) regarding the number of points on a projective curve over a finite field. Namely, if one fixes the degree of the curve, but ...
17
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1
answer
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On the Hasse-Weil L-function of $P^n$
So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a ...
17
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1
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Rational points à la Chabauty-Coleman
I have been trying to learn the method of Chabauty and Coleman to find rational points on curves; I have been reading an exposition by McCallum and Poonen which was pointed out to me by Emerton in ...
17
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0
answers
998
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Automorphic forms and coherent cohomology
Why is it (and what does it mean) that automorphic forms do not contribute in the coherent cohomology of Siegel modular varieties parametrizing abelian varieties of dimension $d>2$ (see section 7 ...
16
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3
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Are there any rational solutions to this equation?
I am not sure if this is an appropriate question, but I was asked this by a colleague today and do not know how to answer it.
1) Are there any rational solutions to the following equation:
$$x^3-8x^2+...
16
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5
answers
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Is every $GL_2(\mathbb{Z}/n\mathbb{Z})$-extension contained in some elliptic curve's torsion field?
I suppose this question could be phrased in terms of Galois representations, but I'm asking it this way.
Let $n>1$ be an integer. If $K$ is a number field with $\operatorname{Gal}(K/\mathbb{Q}) ...
16
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3
answers
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Tower of moduli spaces in Scholze's theory
My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...
16
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1
answer
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Kapranov's analogies
I just wonder about Kapranov's "Analogies between Langlands Correspondence and topological QFT". I would like to read a more detailed exposition and how one turns that analogy into concrete ...
16
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4
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K3 surfaces with good reduction away from finitely many places
Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...
16
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2
answers
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why we need rigid geometry?
Hello, everyone.
I want to ask some questions about rigid geometry.
1.what is the motivation of rigid geometry?
2.what is the applications of rigid geometry for solving arithmetic problems, ...
16
votes
1
answer
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L-functions and higher-dimensional Eichler-Shimura relation
From what I have been reading I understand that it is a part of the Langlands program to express Zeta-function of a Shimura variety associated to the group G in terms of L-functions attached to G. I ...
16
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2
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Period rings for Galois representations
I have some questions concerning period rings for Galois representations.
First, consider the case of $p$-adic representations of the absolute Galois group $G_K$, where $K$ denote a $p$-adic field. ...
16
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2
answers
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Why does Tate's conjecture imply semisimplicity of crystalline Frobenius?
I'm trying to find a reference for the following fact:
If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}_p$, then the Frobenius endomorphism on the crystalline ...
16
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1
answer
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Smooth proper variety over $\mathbb Q$ with everywhere bad reduction
$\newcommand{\Spec}{\operatorname{Spec}}$
Cross-post from Math.SE, hopefully people more knowledgeable in the field will see the question here on MO.
It is a well-known fact that a smooth projective ...
16
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2
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Good introductory references on moduli (stacks), for arithmetic objects
I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and ...
16
votes
2
answers
489
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Number of height-limited rational points on a circle
Consider origin-centered circles $C(r)$ of radius $r \le 1$.
I am seeking to learn how many rational points might lie on $C(r)$,
where each rational point coordinate has height $\le h$.
For example, ...
16
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3
answers
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Is Multilinear Hilbert's tenth problem version undecidable?
A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.
Is there no general purpose algorithm for ...
16
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1
answer
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How fast can we numerically calculate Kloosterman sums?
Define the usual Kloosterman sum by $$S(m,n;c) = \sum_{\substack{x \pmod{c} \\ (x,c) = 1}} e\Big(\frac{mx + n\overline{x}}{c}\Big),$$
where $x \overline{x} \equiv 1 \pmod{c}$, and $e(x) = e^{2 \pi i x}...
16
votes
1
answer
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What is the automorphic interpretation of the Weil conjectures over finite fields
I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any ...
16
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1
answer
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Applications of $p$-adic Hodge theory
I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...
16
votes
1
answer
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GAGA for henselian schemes
In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a ...