All Questions
Tagged with nt.number-theory ag.algebraic-geometry
196 questions
14
votes
4
answers
4k
views
Deligne's letter to Piatetskii-Shapiro from 1973
Could anyone point me to a place where I could find Deligne's letter to Piatetskii-Shapiro from 1973? It is cited for example in Berkovich's "Vanishing cycles for formal schemes II".
14
votes
1
answer
1k
views
Geometry for Anderson's motives?
Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category ...
- 4,015
14
votes
1
answer
1k
views
Elliptic curves and connected components
Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.
- 345
14
votes
3
answers
2k
views
"Nice" definition of discriminant as alluded to in an answer of Qing Liu
In his answer
here
Qing Liu mentioned "the 'discriminant' of X which measures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R."
Could ...
- 646
14
votes
3
answers
4k
views
Recent progress toward Birch and Swinnerton-Dyer conjecture
Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
- 141
14
votes
1
answer
5k
views
Why are Galois Representations so important in Number theory ?
Dear everyone,
Motivation :
From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have ...
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
14
votes
1
answer
1k
views
A quantitative version of Hensel's Lemma
I've been reading some papers on Igusa zeta functions, and they seem to be implicitly using a "quantitative version" of Hensel's Lemma, which also asserts the number of lifts of a $\mathbb{Z}/p\mathbb{...
- 21.3k
14
votes
3
answers
2k
views
A question on K_1 of an elliptic curve
Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , \...
- 5,637
13
votes
1
answer
771
views
Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's Esquisse he claims that the action of
$$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\...
- 3,799
13
votes
1
answer
973
views
Which degree does a motivic Galois representation show up in?
Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...
- 148k
13
votes
1
answer
1k
views
What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?
Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian ...
- 2,270
13
votes
0
answers
663
views
On a kind of Hilbert irreducibility theorem
Let us work over a number field $k$. Let $C$ be a non-empty open subscheme of $\mathbb{P}^{1}_{k}$, and $X\to C$ a family of smooth, projective hyperbolic curves such that $X(k)\to C(k)$ is surjective....
- 1,000
13
votes
0
answers
501
views
Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
- 6,622
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
12
votes
5
answers
2k
views
Clarification on the weak BSD conjecture
It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...
user85435
12
votes
3
answers
911
views
Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively?
Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that ...
- 531
12
votes
0
answers
676
views
Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
- 913
12
votes
9
answers
6k
views
Proofs of Mordell-Weil theorem
I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to ...
- 14.3k
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
11
votes
2
answers
653
views
Abelian variety with prescribed endomorphism ring
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta_8]$ denote the ring of ...
- 14.2k
11
votes
0
answers
491
views
Can an abelian variety/Q have no non-trivial points over Q_sol?
Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal (pro-)solvable ...
- 11.3k
11
votes
1
answer
1k
views
Equivalence between statements of Hodge conjecture
Dear everyone,
I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has ...
10
votes
1
answer
562
views
Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?
Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...
- 148k
10
votes
1
answer
462
views
Homomorphisms between Oort–Tate group schemes
Let $R$ be a complete local $\mathbf{Z}_p$-algebra, for some prime $p$. In the 1970 paper Group schemes of prime order by Oort and Tate, they write down an explicit finite flat group scheme $G_R(a, b)$...
- 37k
10
votes
3
answers
1k
views
What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
user178596
10
votes
0
answers
541
views
Is the compositum of all quadratic extensions of the rationals an ample field?
Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$
Is there a (geometrically irreducible) smooth variety $V/\mathbb{...
- 11.3k
10
votes
1
answer
719
views
what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?
Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
- 10.7k
9
votes
1
answer
751
views
Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$
In Tate's famous paper about $p$-divisible groups, for a prime number $p$ he asks whether there exists a $p$-divisible group $G$ over $\mathbb Z$ such that $G$ is not a direct sum of $\mu_{p^\infty}$ ...
- 6,622
9
votes
0
answers
2k
views
Exactly Counting the Number of Lattice Points in an $n$-Dimensional Sphere
Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its ...
- 311
9
votes
0
answers
381
views
Explicit construction of the Jacobian of a curve
Let $k$ be an algebraically closed field (of arbitrary characteristic), and $C$ a smooth projective curve over $k$, given by defining equations in projective space. I am looking for an algorithmic ...
- 91
9
votes
1
answer
860
views
Complex manifold defined over $\mathbb{Q}$
If we consider complex projective varieties, to be defined over $\mathbb{Q}$ means that there is a projective embedding whose image is the vanishing locus of a polynomial system with coefficients in $\...
user164740
9
votes
1
answer
2k
views
Overview of Arakelov intersection theory and the Arakelov Chow ring
I'm looking for a reference that gives an overview of the most important properties of Arakelov intersection theory (on arithmetic varieties of arbitrary dimension) and that describes basic properties ...
- 47.4k
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
9
votes
2
answers
1k
views
modularity of algebraic varieties
Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties
and whose L-functions have been shown to be a product of automorphic L-functions?
Thanks.
N
- 2,842
9
votes
0
answers
380
views
How can I "see" that a map is birational?
This came up with the Euler brick.
Let $T=(p,q,r)$ be a Randall triple, i.e. $$(p^2-1)(q^2-1)(r^2-1)=8pqr\ \qquad\text{[eq.1]}.$$ There are tons of maps that map a triple $T$ to another $T'=(p',q',r')$...
- 4,829
8
votes
2
answers
901
views
Forms of algebraic varieties
Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...
- 7,722
8
votes
3
answers
1k
views
Ranks of elliptic curves depend only on the field?
Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?
- 11.3k
8
votes
2
answers
804
views
Field of definition of dominant morphisms
Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes k$...
- 382
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
8
votes
1
answer
753
views
Hasse principle and Brauer-Manin obstruction for forms of large degree
The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the ...
- 26.9k
7
votes
0
answers
666
views
High dimensional analogue of Ramanujan's pi formula
The question below comes to my mind when I am trying to explore something related to the formulas found by Jesus Guillera:
a)Generalized hypergeometric function
$${}_3 F_2\left(\begin{matrix}1/4&...
- 3,337
7
votes
2
answers
639
views
Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
7
votes
3
answers
572
views
Siegel's theorem with real coefficients
Let $h(x,y)$ be a polynomial with real coefficients. Suppose there are infinitely many integer solutions to $|h(x,y)|<1$. What can I say about $h$?
When $h$ itself has integer coefficients, a ...
- 156k
7
votes
3
answers
348
views
The rank of elliptic curves and related quadratic twists
Let $E/\mathbb{Q}$ be an elliptic curve, and let $k_1, k_2$ be square-free integers. Can anything be said about the related elliptic curves
$$\displaystyle E/\mathbb{Q}, E^{(k_1)}/\mathbb{Q}, E^{(k_2)}...
- 26.9k
7
votes
2
answers
2k
views
What is the relationship between the finiteness of the Tate-Shafarevich group and the Tate conjectures?
(I asked this on math-stackexchange, but it seems more appropriate to this forum, so I took it off from there and am posting it here)
After the great answer I got for my previous question about the ...
- 71
7
votes
3
answers
908
views
Do there exist elliptic curves over schemes which have all primes as residue characteristics?
It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every ...
- 10.7k
7
votes
0
answers
944
views
Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres
The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...
- 10.5k
7
votes
1
answer
508
views
What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded
According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. here p. 1.
Assuming ...
- 25.4k
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