The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
1answer
140 views

State of the art in arithmetic moduli of elliptic curves?

In trying to get into the topic of moduli spaces of elliptic curves, the following question arises: What is the state of the art in the topic right now? Deligne and Rapoport describes how the ...
7
votes
1answer
154 views

Dimension of family of hyperelliptic curves

Suppose we have an elliptic curve E with a point $P$ of order $5$ over a field of characteristic $0$. Denote $E'$ the curve $E/\langle P\rangle$. Now let $x$ (resp. $x'$) be a function on $E$ (resp. ...
3
votes
1answer
91 views

Krull's intersection theorem in the q-expansion principle

I'm currently reading the proof of the q-expansion principle in Katz'73 paper "p-adic properties of modular schemes and modular forms" . The principle itself is a Corollary (1.6.2) of Theorem 1.6.1, ...
8
votes
1answer
113 views

Existence of smooth elliptic curves with complex multiplication

this is my first question ever on a platform like this so please forgive me any kind of unintended misbehaving. In Kudla, Rapoport and Yang "On the derivative of an Eisenstein series of weight one" ...
9
votes
1answer
208 views

Primes of ramification index 1 with inseparable residue field extension

I've been reading through Neukirch's Algebraic Number Theory, and I'm a little puzzled about a possibility with ramification of primes. As usual, let $\mathcal{O}_K$ be a Dedekind domain with field ...
5
votes
1answer
70 views

Problems on showing that a reduction map is defined, and that a certain scheme is finite.

I am currently on the last chapters in Liu's book and I am trying to solve the following problem, which is the first step in showing that a certian reduction map is well-defined: Let $X \rightarrow T$ ...
1
vote
1answer
75 views

On an openess property

Let $S=Spec(A)$ where $A$ is a noetherian integral domain. Let $f:X\rightarrow S$ be a flat, proper morphism of schemes. Let $U\subset X$ be an open and $V=f(U)$ (in particular $V$ is open by ...
9
votes
1answer
348 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
4
votes
0answers
101 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq ...
2
votes
1answer
43 views

Extend maps between etale groups

Let $V$ be a discrete valuation ring, $S=\operatorname{Spec}(V)$ and $\eta$ (resp. $s$) be the generic (resp. closed) point of $S$. Let $G$ and $H$ be flat group schemes over $S$ and assume I know ...
3
votes
0answers
59 views

étale cohomology of valuation rings

Let $S$ be the spectrum of a discrete valuation ring (we can assume complete or henselian if necessary). Is it true that the ├ętale cohomology group $H_{et}^2(S,\mathbb{Z})$ is zero?If not in general ...
7
votes
1answer
145 views

Completion along zero section of an elliptic curve.

I am trying to understand the intuition that I should have about the formal group of an elliptic curve. Say that I have an elliptic curve $E\to \text{Spec} R$ for some ring $R$, with section $0\colon ...
14
votes
0answers
214 views

What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference ...
5
votes
1answer
189 views

Degree of a Cartier Divisor under pullback

This is question 7.2.3 in Liu's book Algebraic Geometry and Arithmetic Curves and I have been trying with this for some time now. Let $f:X \rightarrow Y$ be a morphism of Noetherian schemes, and ...
11
votes
0answers
227 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
3
votes
1answer
85 views

Is the height associated to a degree zero divisor always bounded?

Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field, and let $D$ a divisor on $X$. To these data, we can associate a height function on the ...
2
votes
0answers
75 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
8
votes
1answer
122 views

Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free

Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
9
votes
2answers
221 views

Intuitively, what is the height of a point on an abelian variety?

I have been reading through Silverman's classic text on elliptic curves and I just can't seem to wrap my head around the height functions. It just kind of shows up. What exactly does the height ...
6
votes
1answer
226 views

Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf

Let $B$ a smooth projective connected variety over $\mathbf C$. Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero. Does the converse hold? That is, suppose that $B$ ...
7
votes
2answers
184 views

Are endomorphisms of degree one always automorphisms

Let $B$ be a smooth projective connected variety over $\mathbb C$. Let $\sigma:B\to B$ be an endomorphism of degree one. Do I understand correctly that $\sigma$ is an automorphism? I believe this ...
8
votes
1answer
223 views

Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
11
votes
2answers
192 views

Families of curves over number fields

Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
7
votes
1answer
120 views

Do K3 surfaces with an Enriques involution have a polarization of bounded degree

Does there exists a real number $C$ with the following property. For any Enriques surface $E$ over a number field $K$ with K3 cover $X\to E$, there exists an ample divisor $H$ on $X$ such that $H^2 ...
3
votes
1answer
49 views

Does this diagram of Chern classes and push forwards commute

Let $p:Y\to X$ be a birational proper surjective morphism of regular surfaces, and let $D$ be a divisor on $Y$ such that $p(D)$ is a point. Then $p_\ast D =0$ by definition. Is there an easy way to ...
3
votes
1answer
84 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
1
vote
0answers
56 views

Galois invariant of Tate twists

let $k$ be the maximal extension of $\mathbb{Q}$ unramified outside a set $T$ of primes in $\mathbb{Z}$. Take a $p\in T$ and set $G=Gal(k/\mathbb{Q})$. I would like to now if there is a classical ...
8
votes
1answer
373 views

Why $p$-adically interpolate?

I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
3
votes
1answer
89 views

Torsors under elliptic curves splitting over the same fields

I have a question somewhat related to my last question. Suppose $C$ and $C'$ are two genus $1$ curves (smooth, projective, geom conn.) over a perfect field $k$ with no $k$-rational points and that $C$ ...
10
votes
1answer
111 views

If $E/\mathbf Q$ is an elliptic curve and $n$ is odd, then the $n$-torsion $E(\mathbf Q)[n]$ is cyclic; elementary proof?

I know that this follows from the existence and non-degeneracy of the Weil pairing. A consequence of the existence of the Weil pairing is that, if the whole $n$-torsion is defined over $\mathbf Q$, ...
35
votes
3answers
665 views

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
7
votes
1answer
197 views

Picard group of genus one curve

Is there a known example (or at least moral reason why such a thing should exist) of a genus $1$ curve $C/k$ over a field (assume perfect if you want) with no rational points such that ...
2
votes
1answer
50 views

Varieties with infinitely many topological covers of finite degree

Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers. If $\dim X =1$, this holds if and only if the genus of $X$ is positive. Do we have a similar ...
7
votes
2answers
199 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
3
votes
0answers
51 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
7
votes
1answer
135 views

Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
6
votes
1answer
572 views

Definition of tamely ramified

I think I can show that the following definitions of "tamely ramified" coincide. I thought it would be good to be sure. Sorry for the easy questions. Let $O_K$ be a dvr with maximal ideal $\mathfrak ...
9
votes
2answers
208 views

Elliptic curves over Spec Z

I want to show that there are only finitely many elliptic curves over Spec $\mathbf Z$ without appealing to Siegel's theorem or Shafarevich' theorem. Firstly, I think (but I am not sure) that such an ...
3
votes
0answers
53 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
7
votes
1answer
250 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
1
vote
0answers
46 views

glueing formal sheaves to obtain a maximal ideal

consider $S=Spec(\mathbb{C}[t])$ and $C\rightarrow S$ a family of proper curves with $C_{\mathbb{C}[t,t^{-1}]}$ smooth and $C_{t=0}$ nodal given by 2 irreducible components $C_1,C_2$ that intersect ...
1
vote
0answers
48 views

Are there moduli spaces of higher-dimensional varieties

In short, the answer to the question is yes. I'm aware of the existence of moduli spaces for canonically polarized varieties with fixed Hilbert polynomial over $\mathbf C$. I think they require the ...
2
votes
1answer
71 views

Pulling-back a divisor and reducing it

Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$. Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, ...
10
votes
0answers
328 views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
9
votes
1answer
242 views

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
4
votes
1answer
81 views

Why is the rank of $f_\ast L$ the degree of $f$

Let $f:X\to Y$ be a finite morphism of curves. Let $L$ be a line bundle on $X$. Why is $f_\ast L$ a line bundle and is the degree of $f_\ast L$ equal to $\deg f$ or $\deg f+ \deg L$? Here is my ...
4
votes
0answers
138 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
2
votes
1answer
155 views

Abelian subvarieties of a principally polarized abelian variety are principally polarized

Let $A$ be a principally polarized abelian variety. Let $X\subset A$ be an abelian subvariety. Is $X$ also principally polarized? Here's what I think should be a proof. Is it correct? We may and do ...
7
votes
1answer
139 views

Are there infinitely many pairs of rational numbers such that…

Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in ...
4
votes
1answer
185 views

The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...