All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
7
votes
1
answer
489
views
Tate-Shafarevich groups over finitely generated fields
Let $G$ be an algebraic group over a number field $k$. One defines the Tate-Shafarevich set of $G$ to be
$$Ш(k,G) = \ker\left(H^1(k,G) \to \prod_{v} H^1(k_v,G)\right),$$
where the product is over all ...
- 21.3k
3
votes
0
answers
154
views
Examples of subspaces singled out by modular forms
I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature.
To be more clear let me explain the example I have in mind.
Let $N\...
- 231
1
vote
0
answers
119
views
The minimum genus of a family of degree $12$ algebraic curves which comes from the resultant of two quartic polynomials
Let $f(t)$ be a rational normal cubic curve in $\mathbb{P}^3$ (it is not contained in any plane) and also we assume that this cubic curve passes through two points $(0,0,0)$ and $(1,0,0)$. By an easy ...
- 39
9
votes
2
answers
979
views
modular forms, invertible sheaves, and quotients
I'm very confused about some contradicatory statements, and I hope someone can help me clarify this.
Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for $\...
- 597
3
votes
0
answers
94
views
Finding a divisor on a curve in a given linear equivalence class made with points over a "small" field
Suppose I have a curve $C$ defined over $\mathbb Q$ and a line bundle $\mathcal L$ on it,
also defined over $\mathbb Q$. I am interested in trying to find a (non-effective) divisor $D=\sum_i a_ip_i$ ...
- 5,186
5
votes
1
answer
269
views
Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve
Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...
- 7,609
10
votes
1
answer
626
views
K3 surfaces that correspond to rational points of elliptic curves
In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the ...
- 5,186
7
votes
1
answer
841
views
Whats the prime-to-$p$ Etale fundamental group of the affine line minus two points over $\mathbb{F}_p$?
Background:
I've often heard that the fundamental group of $\mathbb{P}^1/\mathbb{Q}-\{0,1,\infty\}$ is extremely hard to understand. First of all, it has a surjective map to the galois group
$\rm{Gal}(...
- 2,824
5
votes
1
answer
508
views
Is the ordinary locus affine?
Let $p$ be a prime number and let $Y$ over $\mathbb F_p$ be a Siegel modular variety, with minimal compactification $X$. It is well known that $X^{\operatorname{ord}}$, the ordinary locus of $X$ is ...
- 77
1
vote
1
answer
187
views
Conjugate surfaces: informations about the orbits
Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...
- 1,237
12
votes
1
answer
1k
views
Motivic L-function vs motivic zeta function
Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of
$$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$
where $Fr_p$ is a ...
- 3,799
10
votes
1
answer
564
views
Newton polygons of modular polynomials
This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that ...
- 5,186
16
votes
1
answer
1k
views
Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$
Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
- 163
12
votes
3
answers
2k
views
Weil's Riemann Hypothesis for dummies?
Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example:
(a) For any projective curve $X$ satisfying certain ...
- 7,633
1
vote
1
answer
1k
views
A letter from J. P. Serre
Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
- 61
5
votes
2
answers
423
views
Time-line until the publicaton of Weil of "Numbers of solutions of equations in finite fields"
In "On the history of the Weil Conjectures" Dieudonné says:
"Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...".
C. F. Gauss, ...
- 61
11
votes
0
answers
264
views
What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?
I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that $H^{...
- 149k
6
votes
1
answer
658
views
Connection of Galois representation and arithmetic geometry
There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that people are interested in studying ...
- 601
5
votes
0
answers
585
views
Bloch Kato Exponential as formal lie group exponential
Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) \...
- 3,555
4
votes
1
answer
287
views
Regular singularities and the infinitesimal site
Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$.
A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent ...
- 806
0
votes
1
answer
138
views
rank of Abelian schemes under ample hypersurface section
Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...
user19475
1
vote
2
answers
2k
views
Non-hyperelliptic curves of genus at least two
A hyperelliptic curve can be understood as the set of points satisfying an equation of the form
$$\displaystyle z^2 = f(x,y),$$
where $f(x,y)$ is a binary form of degree $d = 2g+2$. In this case, $g$ ...
- 26.9k
4
votes
0
answers
188
views
Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure
This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
- 1,838
29
votes
3
answers
2k
views
$\zeta(n)$ as a mixed Tate motive
I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...
- 291
5
votes
1
answer
755
views
Elliptic curve and Galois representation
For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = {\...
- 101
2
votes
0
answers
531
views
vanishing of étale cohomology of affine surface
Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime.
Are there vanishing results for ...
user19475
6
votes
1
answer
1k
views
A weak version of Bass' conjecture
Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...
- 21.3k
2
votes
1
answer
420
views
Some questions related to Iwasawa invariants of elliptic curves
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$.
Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
- 1,209
5
votes
2
answers
1k
views
Local factors of Hasse-Weil zeta function - what do they have in common?
Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
- 5,637
15
votes
1
answer
966
views
Curves on K3 and modular forms
The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...
- 600
21
votes
0
answers
617
views
Bounding failures of the integral Hodge and Tate conjectures
It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
- 21.3k
1
vote
0
answers
331
views
Where can I find the article of A. Borel: "Values of zeta-functions at integers, cohomology and polylogarithms"? [closed]
Where on the internet can I find this article?
I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.
user51486
8
votes
2
answers
833
views
is there a p-adic Borel theorem?
Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The ...
- 81
9
votes
1
answer
781
views
Variant of Hilbert 90 for Galois extensions
Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$.
Obviously, $G\subseteq Aut(K)$. It is well known that
$H^1(G,...
- 3,998
11
votes
1
answer
906
views
Serre's 1987 letter to Tate about mod p modular forms
In what follows, we have a level $N \geq 3$, and the modular curve $X(N)$, and the invertible sheaf $\omega$ on $X(N)$ such that the global sections of $\omega^{\otimes k}$ correspond to modular forms ...
- 597
3
votes
1
answer
437
views
Galois representation attached to $3$-torsion points of an elliptic curve
Let
$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $.
$G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$.
$ E[3] $ - $3$-torsion points ...
- 193
12
votes
1
answer
616
views
What is our current knowledge on the structure of J_0(N)(Q) and J_1(N)(Q)
The question in the title naturally breaks up in two parts, namely the torsion part and the rank part. I already read about some results on both the torsion and the rank part. And I want to know ...
- 2,224
11
votes
0
answers
722
views
Degeneration of etale Hochschild--Serre exact sequence
Let $k$ be a field, $X$ a smooth $K$-variety and $\ell$ a prime not dividing the characteristic of $K$.
Then one can make sense of continuous $\ell$-adic etale cohomology (in the sense of Jannsen), ...
- 37k
5
votes
1
answer
516
views
Reference for Local class field theory via witt vectors
I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...
- 71
10
votes
3
answers
882
views
Quadratic twist of curve defined over finite field
I was reading the paper on "curves of every genus with many points II" by: Elkies, Howe, et al. And some of the terms are not clear to me. Is there any elaborate exposition on these stuffs?
In ...
- 135
3
votes
0
answers
155
views
Lattice with trivial spinor norm
Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows:
For a ...
- 31
9
votes
1
answer
633
views
Motives over finite field not generated by hyperelliptic curves
So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.
p.s. A ...
- 1,856
4
votes
2
answers
365
views
Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny
Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ \lambda_{E}^...
- 1,209
27
votes
3
answers
4k
views
Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...)
Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...
- 32.2k
4
votes
1
answer
1k
views
Confusion regarding the definition of semistable reduction of an elliptic curve at a prime $p$
I am consulting the recent paper ''On the Integrality of Modular Symbols and Kato's Euler system for Elliptic Curves'' by Chris Wuthrich. But I am confused regarding the definition of semistable ...
- 193
14
votes
1
answer
2k
views
The Tate conjecture for abelian varieties
Let $k$ be a number field. Recall that Faltings proved the famous Tate conjecture, which states that for any abelian variety $X$ over $k$ and any prime $\ell$, the natural map
$$\mathrm{End}(X) \...
- 21.3k
5
votes
0
answers
205
views
Local Systems on Function fields over $\mathbb{F}_p$
Suppose $X$ is a smooth proper curve over $\mathbb{F}_p$ for some prime number $p$. Let $l\neq p$ be a prime, and suppose $L$ is a rank 2 local system over $X$ with coefficients in $\mathbb{Z}_l$ such ...
- 2,824
2
votes
1
answer
182
views
Criterion for R-equivalence of two points on cubic surfaces over $\mathbb{Q}$
The definition of R-equivalence is given in the paper as Definition 4.1. Coarsely speaking, given a field $K$ and a cubic surface over $K$, two points $x,y$ are R-equivalent over $K$ if they can be ...
- 3,337
14
votes
0
answers
1k
views
Lifting Abelian Varieties to p-adic fields
Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
- 1,453
9
votes
4
answers
752
views
Is any quadric birational to a product of Brauer-Severi varieties?
Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let
$$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$
be a non-singular ...
- 21.3k