All Questions
663 questions with no upvoted or accepted answers
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
4
votes
0
answers
130
views
Castelnuovo–Mumford regularity and wedge powers in positive characterisitc
A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if
$$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$.
It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
- 2,356
4
votes
0
answers
213
views
Computing homology class of curve in product of elliptic curves
I have a smooth, projective curve $X/\mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = \prod_{i=1}^g E_i$. Since $H_*(A; \mathbb{Z})$ with the Pontryagin product is isomorphic to ...
- 1,856
4
votes
0
answers
210
views
Frey's elliptic curve and perfect numbers?
Let $E_n:y^2=x(x-\sigma(n)/2)(x+\sigma(n)/n)$ be a Frey-elliptic curve, where $\sigma$ denotes the sum of divisors of the natural number $n$.
If $n$ is a perfect number ($\sigma(n)=2n$) then the $j$-...
- 213
4
votes
0
answers
506
views
Euler Systems and Coleman’s Conjecture
I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
- 99
4
votes
0
answers
88
views
Is there a way to calculate the Picard $\mathbb{F}_q$-number of an (rational or K3) elliptic surface?
Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface
$$
\mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6,
$$
where $a_i(t) \in \mathbb{F}_{q}[t]$. Is there a way ...
- 1,833
4
votes
0
answers
169
views
Extensions of fraction field and residue field
Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$?
This question should be easy ...
- 393
4
votes
0
answers
163
views
References for the computation of the Mordell-Weil group of an elliptic curve
I am reading about the Mordell-Weil group of an elliptic curve over a number field using primarily Silverman's AEC. While the book is excellent in discussing materials prior to Chapter 8, I think ...
- 401
4
votes
0
answers
271
views
Understanding this example of projective geometry in Algebraic matroids
In this paper by Evans and Hrushovski: Projective planes in algebraically closed fields,
they characterize projective planes in algebraically closed fields. These are coordinated by the skew-fields ...
- 407
4
votes
0
answers
541
views
Formula for Neron Severi group of product surfaces
Let $A=E\times E'$ be a surface which is a product of two elliptic curves. Then it is claimed that there is an isomorphism:
$$\mathbb Z \oplus {\rm Hom}(E, E')\oplus \mathbb Z \to {\rm NS}(A)$$ ...
- 3,472
4
votes
0
answers
96
views
Monogenic cubic rings and elliptic curves
By an elliptic curve over $\mathbb{Q}$, we mean a genus 1 curve with a $\mathbb{Q}$-point. By a monogenic cubic order, we mean a unital cubic ring $R$ isomorphic to $\mathbb{Z}^3$ as a $\mathbb{Z}$-...
- 27k
4
votes
0
answers
189
views
How to find a CM point with the image in the elliptic curve under modular parametrization given
everyone! Let $E:y^2+y=x^3-61$ be the minimal model of the elliptic curve 243b. How can I find the CM point $\tau$ in $X_0(243)$ such that $\tau$ maps to the point $(3\sqrt[3]{3},4)$ under the modular ...
- 390
4
votes
0
answers
133
views
Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic
This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site.
Let $C$ be an algebraically closed field of ...
- 41
4
votes
0
answers
169
views
Fibered surfaces degenerating to Frobenius
Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
- 599
4
votes
0
answers
163
views
Regularity of the modular curves $Y(N)$, $Y_1(N)$
I'm reading the five chapter of the book of Katz-Mazur, Arithmetic moduli of elliptic curves, concerning regularity of the moduli problems of $\Gamma(N)$-structures, $\Gamma_1(N)$-structures and $\...
- 133
4
votes
0
answers
259
views
Second derivative at 1 of L function of elliptic curve
Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that
$$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$
where $\gamma$ is Euler's ...
- 13.1k
4
votes
0
answers
194
views
Geometric interpretation of the rationality of the $j$-invariant
Consider the modular curve $X_0(N)$. Let $\Phi_N(X,Y)$ be the modular equation. Then the curve $\Phi_N(X,Y) = 0$ can be interpreted as a model for $ X_0(N)$ because the function field of $X_0(N)$ is $\...
- 2,375
4
votes
0
answers
530
views
Galois representation of an elliptic curve with CM
Let $ E $ be an elliptic curve with complex multiplication by an order $ \mathcal O$ in an imaginary quadratic field $ K $. Suppose that $ E $ is defined over $\mathbb Q(j(\mathcal O))$. Let $n$ be an ...
- 2,375
4
votes
0
answers
220
views
Calculating some Galois cohomology
Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
- 1,283
4
votes
0
answers
412
views
A website which explains Mazur's torsion point theorem
I'm about to read Mazur's paper "Modular curves and the Eisenstein ideal".
It's so long and difficult for me, but I found a website which shows the Mazur's theorem.
This is very short and very very ...
- 1,364
4
votes
0
answers
275
views
Symmetric power contained in tensor power?
Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be ...
- 2,519
4
votes
0
answers
8k
views
Which proportion of elliptic curves over Q is proved to satisfy BSD conjecture?
In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over $ \mathbb{Q} $ satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article ...
- 6,982
4
votes
0
answers
468
views
Quaternion algebras in characteristic 2
Let $k$ be a field and let $Q$ be a quaternion algebra over $k$.
It is well known that, if $\mathrm{char}\,k\neq 2$, one can define $Q$ as the $k$-algebra of dimension $4$ generated by elements $x,y$ ...
- 375
4
votes
0
answers
313
views
Action of the Picard Scheme of an Elliptic Fibration
Suppose that we have a surface $X$ defined over a field $k$ (I am interested in $k$ being a number field) and an elliptic fibration $f: X \rightarrow \mathbb{P}^1$, i.e. $f$ is proper and almost all ...
- 241
4
votes
0
answers
369
views
Weierstrass model of an elliptic curve: a line bundle over the base
Let $S$ be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map $\pi : S \to C$ where $C$ is a compact Riemann surface.
...
- 587
4
votes
0
answers
122
views
Finding short linear combinations in abelian groups
Let $M$ be a finitely generated abelian group. Assume we are given a presentation of $M$, that is
\begin{equation*}
M = \frac{\bigoplus_{i=1}^r \mathbf{Z}g_i}{\sum_{j=1}^s \mathbf{Z} r_j}
\end{...
- 20.8k
4
votes
0
answers
159
views
identifying components of points on elliptic curves with Kodaira symbol $I_{2n}^{*}$
Let $K$ be a local field that is complete with respect to a discrete valuation.
When an elliptic curve, $E/K$, has reduction type represented by the Kodaira symbol $I_{2n}^{*}$, its component group ...
- 41
4
votes
0
answers
217
views
Finding Rational Curves on a Surface
Let the field of rational numbers be our base field $k$. I hope to find all rational curves on the following surface $S$ defined by $f$. You can find the motivation in the end.
$f= (x^2y^2)z^3 + (5x^...
- 417
4
votes
0
answers
252
views
Expansion of Jacobi theta function at $p$-torsion
I am aware of the formula $$\Theta(z,q)=z\exp\left( -2\sum_{k\geq 1} \frac{z^{2k}}{(2k)!}E_{2k}(q)\right)$$ for the Jacobi theta function at the origin $z=0$. The definition I am using for the theta ...
- 113
4
votes
0
answers
162
views
sieving in function fields
When we want to find a high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$, we give some scores for the curves (the curves with bigger points on them over various primes in ...
- 131
4
votes
0
answers
145
views
About the main theorem of CM for elliptic curves
The classical main theorem of CM by Shimura states, among other things, that:
Consider a quadratic imaginary $K/\mathbb Q$ and an elliptic curve $E= \mathbb C / \Lambda$ with CM s.t. $End(E)\otimes \...
- 41
4
votes
0
answers
252
views
Height pairings of Heegner points of nontrivial conductor
I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:
(1.) Finding a suitable ...
- 1,805
4
votes
0
answers
302
views
What is the Artin invariant of an elliptic supersingular K3 surface?
Let $X$ be a supersingular K3 surface over an algebraically closed field $k$ of positive characteristic $\!p$. Artin proved in the paper https://eudml.org/doc/81948 that the determinant $\mathrm{disc}(...
- 1,833
4
votes
0
answers
308
views
Complex multiplication and ray class fields
This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
- 41
4
votes
0
answers
285
views
Application of Frobenius splitting in characteristic $0$
In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field.
I am reading Brion and Kumar's book and I can see that there are geometric results can be ...
- 849
4
votes
0
answers
339
views
Minimal discriminant of an elliptic curve in terms of its Galois representation
From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...
- 871
4
votes
0
answers
362
views
Result of Deuring, intuitive way to see it's true/quickest way to prove?
There is the following result of Deuring that goes as follows:
Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...
user61522
4
votes
0
answers
348
views
Example of a genus-1 degree-7 plane curve
I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1.
One can show that for a general ...
- 326
4
votes
0
answers
438
views
When an elliptic curve is a quotient of $\mathbb{G}_a$?
I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$.
When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow e^{...
- 2,227
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
4
votes
0
answers
180
views
minimal conductors among elliptic curves with a fixed CM type
Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...
- 7,609
4
votes
0
answers
185
views
Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?
This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in \mathbb{Z}/3[[x]]$...
- 5,422
4
votes
0
answers
164
views
Is there an analogue of distributions in characteristic p?
Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...
- 1,488
4
votes
0
answers
242
views
Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$
As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion $E_{\ell,2}(q)=E_{2}(q)-\...
- 1,537
4
votes
0
answers
189
views
Does this space of mod 2 modular forms admit a (Z/8)* degree decomposition?
Fix an odd N>0. Let M consist of all odd elements of Z/2[[x]] that are the mod 2 reductions of elements of Z[[x]] arising as the Fourier expansions of modular forms for (Gamma_0)(N); it's easy to see ...
- 5,422
4
votes
0
answers
814
views
Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
- 165
4
votes
0
answers
136
views
A subring of the Serre Swinnerton -Dyer ring of level N modular power series
Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of (Z/...
- 5,422
4
votes
0
answers
401
views
Abelian cubic extensions of Q[i],
Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ ...
- 818
4
votes
0
answers
391
views
What is the status of the equidistribution root numbers of elliptic curves' L-functions
In Section 7 of Alice Silverberg's Rank "Cheat Sheet", Silverberg stated
The Bhargava Conjecture: For each $n >
> 1$ the average size of
$S_{n}(E/\mathbb{Q})$ is
$\displaystyle\sum\...
- 7,072
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 4,902