Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

106 questions
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Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers. Q1. Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...
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Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points. I am interested to know why Mordell and ...
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Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...
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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 ...
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What are “perfectoid spaces”?

This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them? Edit: A bit ...
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I have read Hartshorne's Algebraic Geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know how to use scheme ...
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What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
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A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives

The question below is the follow-up of this question on MathOverflow. Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...
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Existence of fine moduli space for curves and elliptic curves

For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...
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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 ...
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Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense: Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
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Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
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unboundedness of number of integral points on elliptic curves?

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...
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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 ...
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Some arithmetic terminology: “universal domain”, “specialization”, “Chow point”

As a non-connoisseur of arithmetic and arithmetic geometry, I would like to ask about some terminology, which meaning I haven't been able to find out on some books, nor on wikipedia, nor by google. ...
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reduction of CM elliptic curves

Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$: (i) $p$ is inert in End($E$) (ii) $E_p$ is supersingular (iii) The trace of the Frobenius at $p$ is $0$ [...
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Crux of Dwork's proof of rationality of the zeta function?

As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
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Why it is difficult to define cohomology groups in Arakelov theory?

I have been puzzled by the following Faltings' remark in his paper Calculus on arithemetic surfaces (page 394) for a few months. He says: If $D$ is a divisor on $X$, we would like to define a ...
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transcendence of canonical heights

Are there known examples of rational points on elliptic curves/abelian varieties over number fields with transcendental canonical height? Thanks.
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Learning path for the proof of the Weil Conjectures

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
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Have there been any updates on Mochizuki's proposed proof of the abc conjecture?

In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known ...
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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 ...
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Crystalline cohomology of abelian varieties

I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...
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“Fermat's last theorem” and anabelian geometry?

Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...
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Langlands in dimension 2: the Yoshida conjecture

Background: One prominent part of the Langlands program is the conjecture that all motives are automorphic. It is of interest to consider special cases that are more precise, if less sweeping. ...
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Integer points (very naive question)

Well, I don't have any notion of arithmetic geometry, but I would like to understand what arithmetic geometers mean when they say "integer point of a variety/scheme $X$" (like e.g. in "integer points ...
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$p$-adic Hodge Theory for rigid spaces, after P. Scholze

I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties. This question is around the "Poincaré Lemma" in the paper. Throughout, let $X$ be a proper smooth rigid ...
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De Rham cohomology of formal groups

Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be ...
Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(... 5answers 4k views Examples and intuition for arithmetic schemes How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ... 5answers 2k views The significance of modularity for all Galois representations On pg. 1 of the slides of a talk, Henri Darmon wrote: Question: What is an interesting Diophantine equation? A “working definition”. A Diophantine equation is interesting if it reveals or ... 2answers 733 views The Sylvester-Gallai theorem over$p$-adic fields The famous Sylvester-Gallai theorem states that for any finite set$X$of points in the plane$\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of$X$. What ... 1answer 1k views Does smooth and proper over$\mathbb Z$imply rational? Does smooth and proper over$\mathbb Z$imply rational? I think someone told me that this is a standard conjecture. Is it a widely held? held at all? Did someone in particular make this conjecture? ... 4answers 915 views Everywhere locally isomorphic abelian varieties Is there a standard example of two abelian varieties$A$,$B$over some number field$k$which are$k_v$-isomorphic for every place$v$of$k$but not$k$-isomorphic ? 3answers 2k views Understanding Faltings's Theorem I am soon to become a graduate student and so I started a personal project; I want to understand Faltings's proof of the Mordell conjecture. I want to get into arithmetic geometry (since I always ... 2answers 2k views Period rings for Galois representations I have some questions concerning period rings for Galois representations. First, consider the case of$p$-adic representations of the absolute Galois group$G_K$, where$K$denote a$p$-adic field. ... 1answer 1k views What geometric properties do properties of ell-adic Galois representations imply? This is the converse question to an earlier question. More precisely, Let$X/K$be a curve(or variety) over a global field$K$. We consider the Galois representation obtained by the absolute Galois ... 0answers 336 views Artin L-function and Zeta function of twisted Dirac operator If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation$\sigma$as a flat connection on the curve, then the definition of the Artin L-... 0answers 293 views Extension of Messing-Mazur-Oda to general groups The following may be well-known (or obviously false), but I can't find a counterexample or a reference. Suppose that$k$is some perfect field (one can assume algebraically closed, if that makes you ... 1answer 1k views Is the group of integer points on a finite-type group scheme over Z finitely presented? Let$G$be a group scheme of finite type over$\mathbf{Z}$. Must$G(\mathbf{Z})$be finitely presented? (The question is inspired by a not yet successful attempt to answer a question of Brian Conrad.... 0answers 717 views Smooth proper schemes over Z with points everywhere locally This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question. Question. Is there a smooth proper scheme$X\to\operatorname{...
By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$. Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...