# Tagged Questions

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

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### Galois cohomology of $GL_n(E^s \hat{\otimes} R)$

Let $E= \mathbb{F}_p(\!(u)\!)$ and write $E^s$ for a separable closure. Write $G_E = \mathrm{Gal}(E^s/E)$ for the absolute galois group of $E$. Let $R$ be a noetherian $\mathbb{F}_p$-algebra and write ...
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### Fields generated by torsion points of CM elliptic curves

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication). I think there is a mistake in his Corollary 1.7 and I'm ...
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### Drawing graph with some different metric values [closed]

I have some metrics but all of them have different values. Some are decimals, some are integers and some are large numbers. Assume the metrics are (average values): - metric1 - 1500 - metric2 - 0....
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### Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?

I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory. As I understand it, there is a very special class of ...
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I'm learning Arakelov theory on arithmetic surfaces and I have the following general question. Let $K$ be a number field and consider its ring of integers $O_K$. Moreover let $S:=\operatorname{Spec} ... 0answers 176 views ### Equivalence of algebraic and topological monodromy representations? Does anyone know of a reference for the following fact? Let$M_g$denote the moduli stack of genus g curves, let$A_g$denote the moduli stack of abelian varieties, and let$U_g \rightarrow A_g$... 0answers 793 views ### Recent progress on the verification of Mochizukis proof of the abc conjecture? [closed] Apparently in preparation for the upcoming workshop on "Interuniversal Teichmüller Theory" in Kyoto in two weeks, which is intended to bring more light into Mochizukis proposed proof of the$abc$... 0answers 145 views ### Fiber at infinity of an arithmetic surface$X$as an element of$\widehat{\operatorname{Div}(X)}$Introduction: Let$M$be a Riemann surface, then a Green function on$M$is an element$g\in C^\infty(V)$where$V=M\setminus\{x_1,\ldots,x_r\}$and around each point$p\in M$we have: $$g=a\log\... 1answer 315 views ### Rational curves on the Fermat quartic surface Let X be the Fermat quartic x^4+y^4+z^4+w^4=0 in \mathbb P^3. It is known that X contains infinitely many (-2)-curves, that is, smooth rational curves. (One way to obtain in infinitely many ... 1answer 294 views ### Nonabelian H^2 and Galois descent I would like to know whether the following metatheorem on nonabelian H^2 has been ever stated and/or proved. Let k be a perfect field and k^s its fixed separable closure. Let X^s be a variety ... 1answer 295 views ### Polylogarithm sheaves In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: \mathrm{... 1answer 279 views ### Geometric and arithmetic Frobenius I read in Serre's "Lectures on N_X(p)" that when X is a scheme defined over \mathbb{F}_q (a finite field), the geometric Frobenius F: X \mapsto X is defined by fixing every element of the ... 1answer 264 views ### Which of these 4 definitions of Galois coverings of integral schemes are equivalent? Here are four possible definitions for an etale, finite, surjective map X\rightarrow Y between integral schemes to be considered Galois: There exists a finite group G, and an action \varphi: G\... 0answers 145 views ### Lifting points via étale morphism of adic spaces This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ... 1answer 67 views ### Bilinearity of the Cassels-Tate pairing Let K be a number field and let A be an abelian variety over K (I'm mostly interested in the case that A is an elliptic curve). We use v to denote places of K and we write H^i(k, A) for ... 0answers 106 views ### Is there a reference for boundedness of smooth canonically polarized varieties over Z (No…) In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack \mathcal M_P of smooth canonically polarized varieties over Spec \mathbb Z with ... 1answer 287 views ### Are there analogies between \Bbb F_q[x_1,x_2] and a suitable object related to \Bbb Z? Much progress in understanding \Bbb Z is made from analogies between \Bbb F_q[x] and \Bbb Z. Can there be analogies between arithmetic in \Bbb F_q[x_1,x_2] and a suitable object related to \... 0answers 121 views ### Reduction “modulo p” of \mathfrak{p}-torsion points of CM elliptic curves Let E/L be an elliptic curve defined over a number field L. Assume moreover that E has complex multiplication by an imaginary quadratic field K. Let \mathfrak{p} be a prime ideal of K. ... 0answers 106 views ### Lifting morphisms of p-divisible groups using Grothendieck-Messing theory During my reading of Peter Scholze and Jared Weinstein's paper Moduli of p-divisible groups'' I found this assertion in the proof of Proposition 6.1.3. Consider the following situation. Let k be ... 0answers 88 views ### U_p operator is not compact on p-adic modular forms I know that one of the reasons for introducting overconvergent p-adic modular forms is that the U_p operator is compact on them. Is there an easy way to see that U_p is not compact on non-... 1answer 188 views ### Tate modules of elliptic curves with complex multiplications Let E/K be an elliptic curve with complex multiplication over an imaginary quadratic field K. Then, I heard that it is well-known that the Tate module V_{p}(E) over \mathbb{Q}_{p} ... 0answers 86 views ### Can I combine the category of Drinfeld modules and the category of the base O_S I am learning about Drinfeld modules,T-modules,...They are said to be analogues of elliptic curves, abelian varieties,... Let K be a finite extension of k = Frac(A), and O_K the integral closure of ... 1answer 155 views ### Regular minimal model of X_0(p^2) Consider the compactified modular curve X_0(p^2) and the corresponding algebraic curve over \mathbb{Q}. My questions are the following: Where do the cusps of X_0(p^2)_{\mathbb{Q}} live? That ... 1answer 426 views ### Elements of arbitrary large order in the first Galois cohomology of an elliptic curve Let E be an elliptic curve over k=\mathbb{Q}. Consider H^1(k,E). In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ... 1answer 200 views ### Algebraic points of uniformly bounded degree on an algebraic variety Let k be a perfect field, and let \bar k be a fixed algebraic closure of k. Let \overline{X} be a nonempty smooth algebraic variety over \bar k. Does there exist a natural number d=d(\... 1answer 218 views ### Archimedean fibers “intersecting” curves on arithmetic surfaces Let's fix a number field K with its ring of integers O_K. Moreover consider an arithmetic surface f:S\to \text{Spec } O_K. For every archimedean place \sigma in K, K_\sigma is the ... 0answers 716 views ### Is it worth the efforts to read books/papers written in Weil's algebraic geometry language There is much important work written in Weil's language of algebraic geometry rather than schemes (besides Weil himself, I can think of Shimura, Neron immediately). My question is: is it worth the ... 1answer 245 views ### Average size of p-part of the Tate-Shafarevich group for elliptic curves Let E/\mathbb{Q} be an elliptic curve defined over \mathbb{Q}. For a given prime p, the p-Selmer group \operatorname{Sel}_p(E) of E and the p-part of the Tate-Shafarevich Ш_E[p] group ... 0answers 80 views ### Fastest algorithm to compute isogeny Let E/GF(p) and E'/GF(p') are two isogenous elliptic curves(\#E=\#E'). We know that there exist the map$$\psi : E \to E'$$Suppose that we haven't any information about degree of \psi. ... 1answer 293 views ### Morphisms for good reduction are maps respecting filtration Please see edits below! So, let A,A'/K be abelian varieties where K is a p-adic local field with residue field k. Suppose further that they have good reduction with models \mathscr{A},\... 0answers 75 views ### Split multiplicative galois representation and specialization My questions stems from my attempt to understand the paper of Greenberg and Stevens about the Mazur-tate-Teitelbaum conjecture (you can find the paper here). To understand this question you probably ... 0answers 180 views ### A Hartogs-type criterion for flatness Let U be a smooth affine connected variety over \mathbb C and let V\subset U be an open whose complement is of codimension at least two. Now, let Y be a smooth quasi-affine connected variety ... 0answers 206 views ### Equations for Elliptic Curves An elliptic curve C over a field k is a smooth, genus 1 curve defined over k with an associated k-rational point. If char(k) \ne 2, we can show that C has a model of the form y^2 = f(x) ... 0answers 118 views ### How to Taylor series expand at the prime at infinity Given a rational number, one can find a Taylor series expansion with respect to any p-adic valuation, as covered in Gouvea's introductory text on p-adic numbers. My question is how does one do ... 1answer 138 views ### Congruence Primes and Modular Degrees Let \mathcal{S}=S_2(\Gamma_0(N) \cap \mathbf{Z} [[ q ]] be the set of cusp forms of weight 2 on \Gamma_0(N) with integral coefficients. Let f \in \mathcal{S} be a normalized newform, so it ... 0answers 81 views ### Reference request: "effective'' semistable reduction I am looking for the origin of the following idea: suppose m and n are relatively prime integers \geq 3. Let E be an elliptic curve over a number field K. Let L/K be a finite extension ... 0answers 80 views ### Intersection of modular polynomial roots Let l,l' and p be three distinct prime numbers and \Phi_k(X,Y) is k-th modular polynomial defined over GF(p). Suppose that we know \Phi_l(X,j) and \Phi_{l'}(X,j) have two roots. Is this ... 5answers 1k views ### How much do I need to learn algebraic geometry to understand arithmetics over number fields I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ... 0answers 295 views ### 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 ... 0answers 94 views ### Weil restriction of fiber products Let$X,Y,Z$be smooth geometrically integral proper varieties over a field$K$where$K/k$is a finite extension of a number field$k$. Let$R|_{K/k}$denote the Weil restriction. Suppose we have$K$-... 1answer 114 views ### Is torsion submodule of a$p$-adically complete and separated$\mathbb{Z}_{p}$-module closed? I was asking to myself the following question. Consider a$p$-adically complete and separated topological algebra$R$over$\mathbb{Z}_{p}$. As$\mathbb{Z}_{p}$is a domain, we know that the$\mathbb{...
If $X$ is a scheme over, let's say, $\mathbb{Z}_p$, one can consider its special fiber obtained by reduction modulo $p$ ans it certainly makes sense to ask if this special fiber is smooth or not. ...