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      Questions tagged [arithmetic-geometry]

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

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      95 views

      Do many homogeneous polynomials help in faster integer root extraction?

      Given $n$ homogeneous algebraically independent total degree $2$ polynomials with no $x_1^2,\dots,x_n^2$ variable in $\mathbb Z[x_1,\dots,x_n]$ with promise that it has non-zero integer roots with ...
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      votes
      1answer
      83 views

      Identifying elements in the kernel of an explicit endomorphism of a Jacobian variety

      I hope this question fits here. Let $H/k$ be a genus $2$ curve and $J$ its Jacobian variety. Since $J(k)\cong \text{Pic}^0(H)(k)$ we have that its generic point looks like $[(x_1,y_1)+(x_2,y_2)-2\...
      1
      vote
      0answers
      48 views

      Unique way to topologise finite algebra over Huber ring

      Let me start with the following Lemma. $\textbf{Lemma}$ Let $A$ be a Tate ring, and let $f\colon A\to B$ be a finite $A$-algebra. Then there is a unique way to topologise $B$ turning it into a ...
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      votes
      0answers
      56 views

      Current status of uniform boundness of rational points on higher genus curves

      We know Lang conjecture can imply uniform boundless of rational points on higher genus (smooth projective) curves over a fixed number field by works of Mazur and others. How is the conjecture of ...
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      vote
      0answers
      84 views

      Maximum number of bounded primitive integer points in a zero-dimensional system

      Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
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      vote
      0answers
      59 views

      What is the probability of 'yes' to this likely $coNP$ problem?

      Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$. Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...
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      vote
      0answers
      60 views

      Heights for rational points via Neron models

      I only just started reading about heights in arithmetic geometry, so forgive me if this question is naive. Suppose $E$ is an elliptic curve over a number field $K$ with ring of integers $R$ and let $\...
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      5answers
      1k views

      Connection Between Knot Theory and Number Theory

      Is there any connection between knot theory and number theory in any aspects? Does anybody know any book that is about knot theory and number theory?
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      1answer
      230 views

      Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$

      My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular: Given a projective variety $V$ defined over $L$ algebraically closed, of ...
      3
      votes
      1answer
      218 views

      Extending section of étale morphism of adic spaces

      This question is related to Lifting points via étale morphism of adic spaces. Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...
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      vote
      1answer
      179 views

      On a refinement of Mordell's conjecture for curves

      Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \...
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      votes
      2answers
      304 views

      Down to earth, intuition behind a Anabelian group [closed]

      An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center. I would like to know ...
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      votes
      1answer
      211 views

      The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

      Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$. I know that to construct the Jacobian variety associated to $C$, one ...
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      vote
      0answers
      69 views

      Igusa curve at infinite level

      In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....
      4
      votes
      1answer
      243 views

      Smooth proper variety over a number field with prescribed bad reductions

      Given a number field $K$, and a finite set $S$ containing finite places of $K$. When can we find a smooth proper geomerically connected variety $X$ over $K$ such that $X$ has good reduction outside $S$...

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