# Questions tagged [arithmetic-geometry]

Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

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### Polynomial values are powers of two

The initial question comes from Komal in 1999. Namely it asks to show that for infinitely many $n$ there is a polynomial $f\in\mathbb{Q}[X]$ of degree $n$ such that $f(0),f(1),\dotsc,f(n+1)$ are ...
1 vote
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### Singularities of arithmetic surface

I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves. The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
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Let $C\subset\mathbb{C}^2$ be an irreducible algebraic curve defined over a number field $F.$ Suppose that for any $(z, w)\in C, z\in \mathbb{\overline{Q}}, w\in\mathbb{\overline{Q}},$ either $z\in F(... 1 vote 0 answers 80 views ### Image of the Kummer map for abelian varieties over$p$-adic local fields The following statement might be well-known to the community: let$K$be a finite extension of$\mathbb{Q}_p$for some prime$p$. Let$A$be an abelian variety over$K$. Then the image of the Kummer ... 3 votes 1 answer 407 views ### If we have a nice formula for number of points on a curve over finite fields, can we get some geometric information of the curve from the formula? Let$p$be a prime number and let$q = p^2$. Let$C$be a separated scheme of finite type over$\mathbb F_q$of dimension$1$. If we know that for every$\alpha \in \mathbb Z_{>0}$, "the ... 4 votes 0 answers 76 views ### Newton stratification in the Shimura variety for$\mathrm{GU}(1,n-1)$over a ramified prime Consider the PEL Shimura variety$\mathrm{Sh}_{K}$for$\mathrm{GU}(1,n-1)$, where$K\subset G(\mathbb A_f)$is an open compact subgroup which is small enough, and$G$is the group of unitary ... 2 votes 1 answer 222 views ### Rational points on a special class of surfaces Consider a smooth surface of the following form $$S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3$$ over$\mathbb{Q}$, and set $$U_S = \{t' \in \mathbb{... 7 votes 1 answer 405 views ### Weil height vs Moriwaki height Let X be a projective veriety over a number field. After fixing an embedding into \mathbb P^n (i.e. a very ample line bundle L), one can define the Weil height \hat h_{L} by restriction of the ... 4 votes 0 answers 176 views ### Torsionness of the kernel of the pullback map of Picard groups of a normalization map Let X be a (irreducible) projective variety over a number field k, \pi: \tilde X \to X be its normalization, and \pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X) be the corresponding map of ... -1 votes 0 answers 93 views ### Is there an superpolynomial integral point degree 2 family satisfying Coppersmith's bounds? Is there an irreducible degree 2 bivariate curve (so of genus 0) which satisfies Coppersmith's bounds but has superpolynomial number of integral points satisfying the bounds (allowed by Falting's ... 8 votes 1 answer 375 views ### Why is the category of motives generated by varieties? I'm reading Ayoub's paper Motifs des varietes analytiques rigides, but I'm not quite familiar with motives. In this paper, he defines the category of motives to be \mathbf{RigDM}^{\rm eff}_{\rm Nis}(... 3 votes 1 answer 155 views ### Stabilizers in abelian varieties are also abelian? reference request Let K be a field of characteristic 0 (number fields is a sufficient generality), A/K an abelian variety, and X\subseteq A a closed reduced subscheme. I am looking for a reference for the ... 3 votes 0 answers 203 views ### Grothendieck trace formula for arbitrary morphisms The Grothendieck trace formula can be viewed as a generalization of the Lefschetz trace formula in étale cohomology from constant sheaves to constructible l-adic sheaves, but restricting to the ... 4 votes 1 answer 240 views ### Rational points of bounded height on a variety I would like to ask for some clarification on the following argument which I can not quite understand. There is a variety X of dimension n over a number field with a degree two map f:X\... 5 votes 1 answer 328 views ### Does p-adic etale cohomology know the variety has ordinary reduction or not? For a smooth proper variety X over discrete valuation ring \mathcal{O} of mixed characteristic (0,p), let X_K be the generic fibre over a generic point \textbf{Spec} K and let X_k be the ... 5 votes 1 answer 243 views ### About closed points in symmetric product schemes over a finite field Let k=\mathbb{F}_q be a finite field with q elements and let X be a quasi-projective k-scheme. I saw somewhere claims the following results (without explanation): Let N be a positive ... 3 votes 1 answer 131 views ### On the exactness of some completed tensor products Let X be an affinoid variety over a discretely valued non-archimedean field k with valuation ring \mathcal{R}. Fix a uniformizer \omega. On the section 3.2 of the paper https://arxiv.org/abs/... 4 votes 1 answer 201 views ### Finding the K=\mathbb{Q}(\sqrt{6})-rational points on the twist of X_{0}(26) Let K=\mathbb{Q}(\sqrt{6}). I am looking to determine all K-rational points on the curve$$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$More precisely, C is a twist of the modular curve X_{0}(... 4 votes 0 answers 215 views ### Number of homomorphisms from a group to \mathrm{GL}_n(\mathbb{F}_q) \DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}Fix a group \Gamma and a positive integer n. Let c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert denote the number of homomorphisms ... 2 votes 0 answers 141 views ### Path spaces vs arc spaces Let X be a smooth projective variety over \mathbf{C} and denote by \mathcal{L}_m(X) the m-th jet space, a smooth \mathbf{C}-scheme representing the functor on \mathbf{C}-algebras$$A\... 3 votes 1 answer 215 views ### Rank of elliptic curves, parity, finiteness of Sha$\newcommand{\Sha}{Ш}\newcommand{\alg}{\mathrm{alg}}\DeclareMathOperator\Sel{Sel}\DeclareMathOperator\rank{rank}$Consider the elliptic curves$E$of$j$-invariant zero that neither them nor their ... 5 votes 1 answer 163 views ### Computation of the torsion of K-groups related to elliptic curves Let$E$be an elliptic curve over$\mathbb Q$. Let$F$be the rational function field of$E$. The$K_2$group of$F$may be described by elements in$F^\times ⊗_\mathbb{Z} F^\times$quotiented by the ... 2 votes 1 answer 384 views ### Galois invariants and tensor products Consider a number field$K$and a finite Galois field extension$L/K$. Let$E$be an elliptic curve over$K$and consider the abelian group $$E(L)\otimes L^{\times}.$$ Every element$g$in$\text{Gal}(... 215 views

If $\{X_i, i\in I\}$ is a directed system of abelian groups such that we have $$\varinjlim_{i\in I}X_i = 0$$ is it true that for every $i$ and large enough $j\ge i$ the transition map $f_{i,j} : X_i\... 1 vote 0 answers 120 views ### On closed subsets in spaces of adèlic points Consider as in Adèlic points and algebraic closure$\mathcal{X}$a projective and flat scheme over$\text{Spec}(\mathcal{O}_K)$, with$\mathcal{O}_K$the ring of integers of a number field$K$. ... 6 votes 1 answer 326 views ### Adèlic points and algebraic closure Consider$\mathcal{X}$a projective and flat scheme over$\text{Spec}(\mathcal{O}_K)$, with$\mathcal{O}_K$the ring of integers of a number field$K$. Let$F/K$vary over all finite Galois number ... 4 votes 1 answer 276 views ### Tate-Shafarevich groups under finite Galois field extensions Suppose$L/F$is a finite Galois extension of number fields. Let$E$be an elliptic curve over$F$and$E_L$its base change to$L$. Do we have that$\text{Sha}(E/F)$is finite if and only if$\text{... 266 views

### Galois cohomology of abelian varieties

Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action. For the first Galois cohomology of $M$, ... 437 views

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### How much results in Calabi-Yau manifolds and mirror symmetry depends on the existence of a ricci-flat metric?

An important result of CY manifold is the CY theorem, it talks about the existence of a ricci-flat metric. However, this theorem and its proof are highly analytic. There are many results about ...
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### Rational points on ramified coverings of abelian varieties

Let $K$ be a number field $A$ an abelian variety over $K$ and $f: X \to A$ a possibly ramified covering of degree $d$ with $X$ a proper variety. My question is: Suppose that $f(X(K)) \neq A(K)$, can ...
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### Simultaneous rank jumping of elliptic curves over number fields

Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...
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### Counting elliptic curves over finite fields with a prescribed number of points

Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$, and let $\mathcal{O}$ be an order in $K$ of discriminant $D$ and class number $h(\mathcal{O})$. Then the Hurwitz-...
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### Is Galois representation induced by semistable elliptic curve semistable?

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Aut{Aut}$Let $E$ be a semi stable elliptic curve. Let $\overline{\rho_\ell}: \Gal(\overline{\Bbb Q}/\Bbb Q)\to \Aut (E[l])$ be mod $\ell$ ...
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1 vote
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### Drawing a 3D object in a 3D environment, and converting to math [closed]

So I have been granted a free time and I want to work on a project but first I had to research. As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
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### Birational geometry over finite fields

I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ... I am studying the following theorem from Silverman's AEC: I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...