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

**2**

votes

**0**answers

53 views

### 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 ...

**2**

votes

**0**answers

139 views

### 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 ...

**4**

votes

**1**answer

147 views

### 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 ...

**-2**

votes

**0**answers

18 views

### 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....

**3**

votes

**1**answer

131 views

### Arakelov divisors and the meaning of real coefficients

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} ...

**5**

votes

**0**answers

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 $U_g \rightarrow A_g$ ...

**53**

votes

**3**answers

18k views

### 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 ...

**10**

votes

**0**answers

770 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 $abc$ ...

**5**

votes

**0**answers

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\...

**7**

votes

**1**answer

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},\...

**12**

votes

**1**answer

314 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 ...

**6**

votes

**1**answer

293 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 ...

**13**

votes

**5**answers

5k views

### Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...

**10**

votes

**1**answer

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{...

**4**

votes

**1**answer

278 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 ...

**5**

votes

**1**answer

263 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\...

**5**

votes

**0**answers

142 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 ...

**3**

votes

**2**answers

809 views

### Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...

**6**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**2**

votes

**1**answer

286 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 $\...

**2**

votes

**0**answers

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 $K$. ...

**4**

votes

**0**answers

105 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 ...

**2**

votes

**0**answers

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-...

**3**

votes

**1**answer

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
$\mathbb{Q}_{p}$ ...

**10**

votes

**1**answer

1k views

### What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...

**4**

votes

**0**answers

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 ...

**5**

votes

**1**answer

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 ...

**0**

votes

**0**answers

143 views

### Is sum $(E_i, E_j)$ non-positive, with $E_i$'s the exceptional components of a desingularization

Let $Y$ be an integral normal 2-dimensional scheme and let $X\longrightarrow S$ be a flat projective morphism, where $S$ is a Dedekind scheme.
Let $f:X\longrightarrow Y$ be a minimal resolution of ...

**11**

votes

**1**answer

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 ...

**10**

votes

**1**answer

311 views

### Integral points on elliptic curves of the form $y^2=x^3+px$

As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be ...

**1**

vote

**1**answer

208 views

### Finding cyclic subgroups of points on elliptic curves for isogeny based cryptography

Isogeny based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is a theorem:
Elliptic curves ...

**8**

votes

**1**answer

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(\...

**11**

votes

**2**answers

445 views

### Equivalence of various definitions of arithmetic Chow groups

If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ ...

**10**

votes

**0**answers

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 ...

**5**

votes

**1**answer

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 ...

**7**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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 $\psi$.
...

**7**

votes

**2**answers

383 views

### Are ranks of Jacobians over number fields unbounded?

Fix a number field $K$.
Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
Does there exist an integer $g$ such ...

**2**

votes

**0**answers

74 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 ...

**7**

votes

**0**answers

1k views

### An example computation of etale cohomology

(edited for clarity)
In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...

**7**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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 $y^2 = f(x)$ ...

**14**

votes

**1**answer

794 views

### 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 ...

**4**

votes

**1**answer

137 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 ...

**3**

votes

**0**answers

117 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 ...

**5**

votes

**0**answers

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 ...

**11**

votes

**5**answers

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 ...

**1**

vote

**0**answers

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 ...

**7**

votes

**1**answer

476 views

### Where was the arithmetic zeta function of a scheme first defined?

Let $X$ be an arithmetic scheme, that is, a scheme of finite type over the integers. We denote the set of closed points of $X$ by $|X|$. For every $x\in|X|$, write $N(x)$ for the cardinality of the ...