All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
2
votes
1
answer
206
views
Density of integral points on affine cubic surfaces of a certain type
Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$.
Is there a general ...
- 26.9k
5
votes
0
answers
147
views
Is the cohomology of Hilbert modular surfaces spanned by special cycles?
We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
- 409
4
votes
1
answer
282
views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
- 353
4
votes
0
answers
221
views
Galois action of Weil restriction
Let $K/\mathbb{Q}$ be a quadratic field. Let $E$ be an elliptic curve defined over $K$ but not over $\mathbb{Q}$, and let $\bar{E}$ be the Galois conjugate of $E$. Then by the descent theory (for ...
- 461
4
votes
1
answer
211
views
anti-holomorphic Hilbert modular forms as global sections
The classical definition of Hilbert modular cuspforms as given in say, Hida's "On $p$-adic Hecke algebras for $\mathrm{GL}_2$ over totally real fields", defines them as holomorphic functions ...
- 302
21
votes
0
answers
2k
views
Recent developments in the proof of Fermat's last theorem
I posted on Mathematics Stack Exchange, but was encouraged to post on MathOverFlow instead.
It has been 20 years since Fermat's last theorem was proved by Andrew Wiles.
Has there been any ...
- 311
28
votes
1
answer
2k
views
SOS polynomials with integer coefficients
A well known theorem of Polya and Szego says that every non-negative univariate polynomial $p(x)$ can be expressed as the sum of exactly two squares: $p(x) = (f(x))^2 + (g(x))^2$ for some $f, g$. ...
- 1,703
2
votes
1
answer
117
views
Effective semi-group of a singular abelian surface
Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous ...
- 1,701
3
votes
0
answers
276
views
Closed form for Jacobi sum $\sum_{a\in \mathbb{F}_{p^2}}\chi(a)\chi(1-a)$
Let $\mathbb{F}_{p^2}$ be a field with $p^2$ elements and $\chi:\mathbb{F}_{p^2}^*\to\mathbb{C}^{*}$
be a multiplicative and non-trivial character on the multiplicative group $\mathbb{F}_{p^2}^*$ (...
6
votes
3
answers
605
views
Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$
Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$
Of course, there are solutions to this like $(a,b,c) = (9,8,6)$.
Is there any known approximation for the ...
- 749
2
votes
1
answer
270
views
Local to global principle for a pair of bilinear equations?
Let $A_{i, j}, B_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations
$$
A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C
$$
$$
B_{1, 1} x_1 y_1 + B_{...
- 3,625
3
votes
0
answers
111
views
Torsion divisor on non-hyperelliptic curve
Let $P$ and $Q$ are two points on a curve $C$. We say $P-Q$ is a torsion divisor if $dP-dQ$ is a principle divisor for some positive integer $d$.
For hyperelliptic curves, Grant proved a Nagell-Lutz ...
- 61
2
votes
1
answer
381
views
Reduced complete Tate ring which is not uniform?
Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
- 217
0
votes
0
answers
219
views
How canonical are integral lifts of Hasse invariants and other mod-$p$ modular forms?
Let $p$ be an odd prime. Recall that the mod $p$ Hasse invariant $A$ of an elliptic curve is an $\mathrm{SL}(2,\mathbb Z)$-modular form of weight $p-1$ defined over $\mathbb{F}_p$. Writing $\overline{\...
- 54.6k
5
votes
0
answers
341
views
Reduction type of elliptic curves over $p$-adic fields and local Langlands correspondence for $GL_2(F)$
In an introductory note of local Langlands correspondence http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf, section $11$ describes a recipe to ...
- 6,622
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
3
votes
1
answer
392
views
Translates of abelian subvarieties
Suppose $A$ is an abelian variety over an algebraically closed field $k$. I'm confused about the notion of translates of abelian subvarieties. I looked over a few related papers, but I couldn't find a ...
- 87
1
vote
0
answers
121
views
Transcendance in function fields
Denote by $\Omega$ the completion of an algebraic closure of $\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$ for the valuation $-\deg$. Let $(a_n)_n$ be a sequence of $\overline{\mathbb F_q(T)}\...
- 3,998
23
votes
2
answers
3k
views
How can I see the relation between shtukas and the Langlands conjecture?
The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.
Drinfeld ...
- 317
9
votes
1
answer
353
views
The $S$-unit equation for functions on curves
Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...
- 20.8k
5
votes
0
answers
354
views
Modern reference for Andre Weil's 'Sur les courbes algébriques et les variétés qui s'en déduisent'
I'm currently interested in the cardinality of the set of values of a polynomial over a finite field.
I found a paper
Saburo Uchiyama, Sur le Nombre des Valeurs Distinctes d'un Polynome a ...
- 1,013
4
votes
0
answers
389
views
Kottwitz global gerbes
I've been trying to understand global gerbes constructed by Kottwitz in $B(G)$ for all local and global fields (arXiv:1401.5728). Scholze explained in $p$-adic geometry (arXiv:1712.03708) why I would ...
- 4,418
2
votes
1
answer
269
views
Perfect square quadratic expression
For a given rational $c\ne-1$, I need to find a rational $x\ne20$ with a small denominator such that $(5cx+100)(5cx-64c+36)$ is a perfect square.
I start with
$y^2=(5cx+100)(5cx-64c+36)$
and ...
- 913
4
votes
1
answer
347
views
Weight 3 modular form associated to singular abelian surfaces?
Given an extremal K3 surface $S$ over $\mathbb{Q}$ (i.e. a K3 surface with maximal Picard rank) there is a 2-dimensional Galois representation on the transcendental lattice $T(S)$, and an associated ...
- 1,701
0
votes
0
answers
165
views
Elliptic curves and archimedean place
here is my question :
Let $K$ be any field, $ E \to spec(K)$. Let $ v \in M_K $ an archimedean place.
We know that $ \overline{K_v} \simeq \mathbb{C}$ and there exists $\tau_v \in \mathbb{H}$ such ...
0
votes
0
answers
82
views
How geometry changes up to Hermitian inner product on Line bundle (Kodaira embedding)
Riemann metric $g \colon= \Sigma g_{ij} dx_i \otimes dx_j$ on a Kähler manifold $M$ will define the length of a line on $M$, i.e. intrinsic geometry. The line bundle $L$ on $M$ is equipped with a ...
- 563
25
votes
1
answer
687
views
Geometry of algebraic curve determined by point counts over all number fields?
Let $C$ be a smooth (geometrically irreducible) projective curve of genus $g>1$ over a number field $K$. The Mordell conjecture (first proved by Faltings) says that for any finite field extension $...
- 2,665
3
votes
0
answers
145
views
Gauss-Manin and Hilbert modular forms
There is a geometric formulation of Hilbert modular forms (HMFs) that parallels that for classical modular forms (sections of a line bundle over the moduli space of Hilbert-Blumenthal Abelian ...
- 949
6
votes
2
answers
926
views
Motivating the coefficient field of $\ell$-adic cohomology
It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures.
It was also ...
- 317
3
votes
0
answers
259
views
Explicit equations for rational elliptic surfaces (Halphen surfaces)
I am looking for explicit equations for rational elliptic surfaces in characteristic $2$. For me, a rational elliptic surface $X$ is a smooth projective surface $X$ which is rational and equipped with ...
- 7,722
3
votes
0
answers
632
views
Bezout's theorem for finite fields [closed]
I would like to know if the following statements holds:
Suppose $K$ is a finite field. Is it true then that for any two polynomials $f,g\in K[x,y]$ that have no common factors, the number of ...
- 14.3k
3
votes
2
answers
338
views
Isomorphism between finite algebras over ${\Bbb Z}_p$
Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$
$R$ is a ...
- 563
3
votes
0
answers
346
views
Flat representations: trying to understand Wiles' proof
Trying to understand the proof of Fermat's last theorem, can anyone explain to me how exactly a finite flat group scheme over $\mathbb{Z}_{p}$ gives rise to a Galois representation over a finite field?...
- 2,125
12
votes
2
answers
758
views
An isogeny between Jacobians of hyperelliptic curves
Let $\mathbf{F}_q$ be a finite field of odd characteristic. Let $X_t$ be the hyperelliptic curve over $\mathbf{F}_{q^2}(t)$ with affine equation
$$y^2 = \left((x^{(q+1)/2}-(x-1)^{(q+1)/2})^2 - t\...
- 4,487
13
votes
2
answers
527
views
Naive point count underestimates the number of mod $p$ points of an elliptic curve for infinitely many primes
Let $E$ be an elliptic curve over $\mathbb{Z}[1/N]$ where $N$ is some non-zero integer. Can one show that that the integer $n_p-p-1$ (where $n_p$ is the number of points of $E$ mod $p$) is positive ...
user145520
1
vote
1
answer
398
views
Boundary divisor of projective toroidal compactification
If $F$ is a totally real number field with $[F:\mathbb{Q}] = d>1$, $X$ is the moduli space of Hilbert-Blumenthal Abelian varieties for $F$, and $\overline{X}$ is the projective toroidal ...
- 949
8
votes
1
answer
692
views
Why can Euler systems constructed from algebraic cycles only be anticyclotomic?
In a footnote to the 2018 Zerbes-Loeffler lecture notes from the Arizona Winter School, it's stated that Euler systems constructed from algebraic cycles "cannot give a full Euler system, only an ...
- 2,044
2
votes
0
answers
100
views
On a certain radical of the formal power series ring $K[[X_1,X_2,\ldots,X_{\infty}]]$
Let $K$ be a field of characteristic $p > 2$ and $A_{\infty} \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be an infinitely-many-variable formal power series ring over $K$ (the symbol $X_{\infty}$ is to ...
- 563
3
votes
1
answer
465
views
Pushforward of functions on a frame bundle
Apologies in advance for the long setup and question.
Let $L \to X$ be a line bundle. We may take its frame bundle $p \colon Fr(L) \to X$, a $\mathbb{G}_m$-torsor. We have
$$ p_*\mathcal{O}_{Fr(L)} =...
- 949
4
votes
0
answers
229
views
Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?
In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. ...
- 2,044
0
votes
0
answers
139
views
Affine surfaces and projective curves
Let $F \in \mathbb{Z}[x_0, x_1, x_2]$ be a geometrically irreducible homogeneous polynomial of degree $d \geq 3$, so that the equation $F = 0$ defines a projective curve $C_F$ in $\mathbb{P}^2$ of ...
- 26.9k
6
votes
0
answers
574
views
Poincaré duality and Galois action
Poincaré duality says that under nice situation we have a canonical perfect pairing
$$ P : H_c^r(X, \mathscr{F}) \times H^{2d - r}(X, \mathscr{F}^\vee(d)) \to \mathbb{Z}/n.$$
I want to show that ...
- 1,364
33
votes
4
answers
6k
views
Is there any theory why (for Bitcoin) the discrete logarithm problem is so hard to solve?
Note I am an active member and contributor at the sister site https://bitcoin.stackexchange.com while studying Bitcoin and as a person who studied mathematics 10 years ago there is one thing I kept ...
- 439
4
votes
0
answers
215
views
Local Tamagawa numbers for algebraic *surfaces*
If $X/\mathbf{F}_q$ were an algebraic curve with closed point $x$, and $G$ a smooth $d$-dimensional affine group scheme over $X$, then the Tamagawa measure $\mu_{\omega, x}$ for the locally compact ...
- 5,701
1
vote
1
answer
241
views
Integral zeros of a multivariate polynomial
Consider the multivariate polynomial
$$f(x_1,\ldots,x_m)=mk\sum_{i=1}^mx_i^2-mk(k-1)\sum_{i=1}^mx_i-\left(\sum_{i=1}^mx_i\right)^2,$$
for integers $m,k\ge2$. We are looking for integral zeros of $f$ ...
- 33
0
votes
0
answers
80
views
Points on hyperelliptic curves coming from an orbit of an algebraic group
Consider a hyperelliptic curve $C_F$ defined over $\mathbb{P}(1,1,g+1)$ by the equation
$$\displaystyle C_F: z^2 = F(x,y),$$
where $F \in \mathbb{Z}[x,y]$ is a non-singular binary form of degree $2g+...
- 26.9k
4
votes
0
answers
222
views
Mordell-Weil group of an abelian variety on the perfect closure of a finitely generated field
This question is closely related the question Over which fields does the Mordell-Weil theorem hold?
I consider the following question:
(1) Let $K$ be a finitely generated field extension of $\...
- 26k
7
votes
1
answer
217
views
Subfields of Hilbertian fields
This question is about the Remark on the top of page 22 of Serre's Topics in Galois Theory, available here :
http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/serre_galois_theory.pdf
My ...
- 353
5
votes
0
answers
459
views
A functor on Abelian varieties corresponding to this operation on Weil numbers
Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$.
Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still ...
- 7,746
1
vote
0
answers
88
views
Algebraic definition of the "pseudo complement" of algebraic curve
Not sure if this makes sense.
Let $K$ be field and $C : f(x,y)=0$ algebraic curve curve over $K$.
Define the "pseudo complement" $\hat{C}$ to be the rational surface
$z f(x,y) - 1=0$ with ...
- 25.4k