All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
6
votes
1
answer
952
views
Is it possible to recover the degree of a field extension from a list of elements and the ground field?
I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
- 1,049
1
vote
1
answer
811
views
Is the direct limit of Weil restriction of an elliptic curve a scheme?
In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very ...
- 4,593
11
votes
0
answers
1k
views
Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
25
votes
2
answers
2k
views
Examples where the analogy between number theory and geometry fails
The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
10
votes
3
answers
1k
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Looking for reference on Serre's talk "linear rep and number of points mod p"
Actually I am not sure this is a legitimate question on MO. In April and June of this year Serre gave two talks on the same title "linear representations and the number of points mod p", one in ETH ...
- 1,160
11
votes
5
answers
2k
views
Analysis of a quadratic diophantine equation
Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, ...
- 113
8
votes
1
answer
2k
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Modular Curves as Moduli Spaces of Elliptic Curves
Hi,
Is the modular curve defined as the quotient of the upper half-plane by an arithmetic group $ \Gamma $ always a moduli space of elliptic curves with extra structure? I know this is true for $ \...
- 309
1
vote
2
answers
307
views
Subset higher power sum question (related to quadratic forms)
Let $\mathbb N_{n} = \{1,2,\cdots,n\}$.
Let $S$ be of cardinality $n$ where elements of $S$ are integers from $\mathbb N_{n}$ and at least one element of $S$ is repeated (That is at least one integer ...
- 13.9k
2
votes
0
answers
380
views
Geometric Inertia Action
Let $K$ be a finite extension of $Q_p$ and $K'/K$ a totally ramified Galois extension
with Galois group $G$. For $g\in G$ and any scheme $X$ over $O_{K'}$, write
$X_g$ for the base change of $X$ ...
- 1,609
9
votes
1
answer
1k
views
Explaining the number field-function field analogy
There is a general circle of ideas according to which true statements about number fields should have analogues in function fields. As best I can tell, the fact that this seems to work is pretty ...
- 159
5
votes
1
answer
970
views
Exponential sums and differential equations
Hi, I have a general question about the relationship between exponential sums and differential equations. In particular, I have been trying to read Katz' work on the subject (his book and his lecture ...
- 661
4
votes
2
answers
442
views
A mapping from a lattice to itself
Consider $\mathbb{Z}^{n}$ for $n = 2^r$ where $r \geq 1$ . Look at the iterates of the following function $T$ from $\mathbb{Z}^n$ to itself.
$T((a_1, a_2, \ldots, a_n)) = (|a_1 - a_n|, |a_2 - a_1|, |...
- 73
8
votes
2
answers
621
views
Geometric decomposition of J(11)
Let $N$ be a prime number. Let $J(N)$ be the jacobian of $X_\mu(N)$, the moduli space of elliptic curves with $E[N]$ symplectically isomorphic to $Z/NZ \times \mu_N$. Over complex numbers we get that $...
- 818
2
votes
1
answer
597
views
Existence question on rational points on a curve
I am puzzled about the following question:
Let C be a smooth, projective, absolutely irreducible curve defined over GF(q) and let g denote the genus of C. O is a rational point on C, and the divisor ...
- 23
22
votes
1
answer
2k
views
Which elliptic curves over totally real fields are modular these days?
As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is ...
- 13.1k
16
votes
2
answers
2k
views
Which languages could appear on Weil's Rosetta Stone?
André Weil's likening his research to the quest to decipher the Rosetta Stone (see this letter to his sister) continues to inspire contemporary mathematicians, such as Edward Frenkel in Gauge Theory ...
- 5,094
5
votes
2
answers
983
views
finite or infinite many quadratic fields embedding into quaternion algebras?
Suppose $H$ is a indefinite quaternion algebra over $\mathbb{Q}$. Are there infinitely many quadratic fields that can be embedded into $H$?
- 709
9
votes
2
answers
533
views
Sum of reciprocals of primes modulo which a polynomial has a root
Dear all,
I am looking for a proof or a reference of the following statement:
Let $f$ be a non-constant polynomial with integer coefficients. Then the sum $\sum \{1/p \mid f \text{ has a root modulo ...
- 95
4
votes
1
answer
308
views
Relation between l-adic and l'-adic geometric monodromy
Suppose $X$ is a smooth family of algebraic varieties over the base $B:=\mathbb{P}^1\backslash\lbrace0,1,\infty\rbrace$ over $\overline{\mathbb{Q}}$; then we can form the relative $l$-adic cohomology ...
- 41
9
votes
1
answer
599
views
Numerical evidence of Beilinson's conjecture in local fields and function fields
The famous Beilinson's conjecture predicts a relationship between the regulator map in $K$-theory and special value of $L$-function generalizing the Dirichlet's theorem in number theory. Please see ...
- 91
9
votes
1
answer
1k
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Isogenies between Tate curves
Let $q$ and $q'$ be complex numbers with $0<|q|,|q'|<1$, and let $m$ and $n$ be positive integers.
Suppose that $q^m={q'}^n$. Then the map
$$
f:\mathbb{C}^\times/q^{\mathbb{Z}} \to \mathbb{C}^\...
- 27.2k
16
votes
4
answers
1k
views
Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?
Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond ...
- 4,593
26
votes
7
answers
6k
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When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of ...
- 4,593
14
votes
1
answer
1k
views
Dirichlet's regulator vs Beilinson's regulator
Consider a number field $F$ with ring of integers $O_F$. The Beilinson regulator can in this particular setting be viewed as a map from $K_n(O_F)$ to a suitable real vector space. Here $n$ is any ...
- 5,637
32
votes
1
answer
2k
views
Structure on $X(k)$ for separated finite type alg. space $X$, for complete valued $k$.
Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$.
If $X$ is a scheme then $X(k)$ inherits a natural (...
5
votes
0
answers
1k
views
Galois groups over function Fields (étale fundamental groups)
Is there a way to compute Galois groups of function field extensions using Pari or Sage? Given a polynomial, $p(x,t) \in \mathbb{F}_p[t][x]$ we can try to solve for $x(t)$, but if no such polynomial ...
- 22.8k
16
votes
1
answer
8k
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How many people fully understand the proof of Fermat's Last Theorem?
What is a rough order of magnitude estimate? $$ $$ There is a thread on Meta about this question, http://mathoverflow.tqft.net/discussion/567/rapid-closing-of-questions/#Item_0
- 99
9
votes
2
answers
983
views
How does one classify finite flat group schemes over a ring where p is nilpotent?
Background: I am trying to work out some Ext calculations for finite flat group schemes over a ring where p is nilpotent. I know how to do these calculations for finite group schemes over a finite ...
- 91
26
votes
5
answers
3k
views
Existence of zero cycles of degree one vs existence of rational points
Let $k$ be a field (I'm mainly interested in the case where $k$ is a number field, however results for other fields would be interesting), and $X$ a smooth projective variety over $k$.
By a zero ...
- 21.3k
1
vote
0
answers
356
views
Quadratic Solutions
There are quadratic solutions to $x^4+y^4 = z^4$ in $\mathbb{Q} (\sqrt{-7})$. But for equations such as $x^4+y^4 = nz^4$ where $n \in \mathbb{N}, \ n \neq 1$ do there still exist extension fields of $\...
- 1
11
votes
0
answers
528
views
Adeles of Holomorphic Functions
In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...
- 15.4k
36
votes
1
answer
9k
views
Fontaine-Mazur for GL_1
For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...
- 21.3k
6
votes
2
answers
754
views
Elliptic curves — general structure of the group
Let $K$ be a field and $E$ be an elliptic curve defined over $K$. It well understood the $K$-points on $E$ forms an abelian group. What is the structure of this group?(Depending on char($K$)?) Is it a ...
- 513
7
votes
2
answers
894
views
Good reduction and blow-ups
Let $X$ be a projective variety over $\mathbb{Z}$, and suppose that $X$ has everywhere good reduction. Let $Y$ be the blow-up of $X$ at an integral point.
Then is it the case that $Y$ also has ...
- 21.3k
29
votes
0
answers
3k
views
What are the possible singular fibers of an elliptic fibration over a higher dimensional base?
An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...
- 3,022
5
votes
4
answers
388
views
Familiar equations in more general settings
What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:
...
- 3,612
8
votes
1
answer
1k
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Monodromy groups of families of abelian varieties: a reference request
In Serre's letter to Vigneras of 2 Oct 1986, he summarizes a course he's giving in Paris, explaining how to control the image of the mod-l Galois representations attached to abelian varieties. In ...
- 19.2k
12
votes
1
answer
566
views
Counting branched covers of the projective line and Spec Z
I've asked a question like this before, but now I'm more interested in counting the number of covers.
We suppose given the following data.
A positive integer $d$
A finite set of closed points $B= (...
- 9,497
11
votes
1
answer
564
views
CM field to Torus to Abelian Variety?
Given a CM field we can use its maximal order (and a choice of CM type) to construct an abelian variety $\mathbb{C}^g/\Lambda$ with complex multiplication by the maximal order.
How do I (or where can ...
- 4,593
7
votes
1
answer
454
views
Checking local solubility of varieties at "bad" primes
Let $X$ be a smooth variety defined over $\mathbb{Q}$. If we want to check that $X$ is locally soluble at a prime $p$, then it suffices to find a non-singular $\mathbb{F}_p$-point, which can be lifted ...
- 21.3k
18
votes
1
answer
6k
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Deligne's proof of Ramanujan's conjecture
I am trying to understand Deligne's proof of the Ramanujan conjecture and more generally how one associates geometric objects (ultimately, motives) to modular forms.
As the first step, which I ...
- 1,783
22
votes
2
answers
8k
views
Geometric vs Arithmetic Frobenius
If an algebraic variety $X$ over a field characteristic p is given by equations $f_i(x_1,...,x_k) = 0$, we can consider the variety $X^{(p)}$ obtained by applying p-th powers to all the coefficients ...
- 1,783
18
votes
4
answers
2k
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Origin of symbol *l* for a prime different from a fixed prime?
I've never seen an authoritative explanation for the choice of the lower case letter $\ell$ or $l$ to denote an arbitrary prime different from a given prime $p$. This now has its own LaTeX command \...
- 52.9k
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 139
7
votes
1
answer
2k
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Local root number
I am reading about the L-functions of elliptic curves and I was thinking about the root number as the product of local root numbers. So my question is how to think about the local root numbers ...
- 995
20
votes
2
answers
2k
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Integral points on varieties
I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.
For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine ...
- 21.3k
3
votes
1
answer
2k
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Looking for an undergraduate research problem in algebraic geometry or algebraic number theory
I'm looking for a small research problem an undergraduate would be capable of after taking just an abstract algebra course, introductory algebraic geometry (at level of Miles Reid's book and Ideals, ...
6
votes
1
answer
395
views
Diameter of reduction graph of a curve over a complete discrete valuation ring
Let $R$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k$, and let $X$ be a proper, smooth, geometrically connected curve over $K$. Take a ...
- 223
-4
votes
2
answers
6k
views
Factorizing polynomials of several variables (in a different perespective)
I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^...
21
votes
2
answers
8k
views
Separable and algebraic closures?
I have no intuition for field theory, so here goes. I know what the algebraic and separable closures of a field are, but I have no feeling of how different (or same!) they could be.
So, what are the ...
- 35.5k