All questions with tag [math: arithmetic-geometry]


The universal cover of the multiplicative group over the field of algebraic numbers

Let $X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$ be the multiplicative over the field of algebraic numbers. Each finite etale cover $Y\to X$ (with $Y$ connected) is isomorphic to the finite etale morphism $X\to X$ given by $z\mapsto z^n$ for some $n\geq 1$. The universal covering space $\widetilde{X}$ of...
2022-07-25 15:45:59

Defining invariants of varieties over fields

Let $f$ be a real-valued function on the set of curves over $\overline{\mathbf{Q}}$. Assume that isomorphism curves give rise to the same value for $f$. Let $K$ be a number field and let $X$ be a curve over $K$. Define $f_K(X)$ to be the value of $f$ after base change to $\overline{\mathbf{Q}}$. Now, a priori, this function $f_K$ is not well-de...
2022-07-24 09:22:05

Why is Hodge more difficult than Tate?

There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow: "[...] we also have a natural abelian (in fact Tannakian) category in play: in the complex case, the category of pure Hodge structures, and [when the field o...
2022-07-24 06:29:48

Flatness of residual representations associated to modular forms

Let $f\in S_k(\Gamma_1(N),\chi)$ be a Hecke eigenform of weight $k\geq 2$, $p$ an odd prime not dividing $N$, and $K_f$ the number field generated by the Hecke eigenvalues of $f$. Fixing a prime $\lambda$ of $K_f$ over $p$, one has the continuous irreducible $p$-adic representation $\rho:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(K_{f,\lambda})$ unram...
2022-07-24 05:26:20

Twists of rational points

Let $X$ be a "nice" algebraic curve over some field $K$, say characteristic zero. The twists of $X$, i.e., the curves $Y$ over $K$ such that $X_{\overline K} \cong Y_{\overline K}$, are in bijection with the continuous cohomology pointed set $$H^1(\mathrm{Gal}(\overline{K}/K),X).$$ I was just wondering about an analogous question for ...
2022-07-22 18:46:01

Is there a fundamental domain for $\Gamma(2)$ contained in the following strip

Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane. Does it have a fundamental domain contained in the strip $$\{x+iy: -1\leq x \leq 1\}?$$ Is the following a correct argument? The matrix $$\left( \begin{matrix} 1 & \pm 2 \\ 0 & 1 \end...
2022-07-20 17:48:41

Can a non-proper variety contain a proper curve

Let $f:X\to S$ be a limited type, apart yet non - correct morphism of schemes. Can there be a projective contour $g:C\to S$ and also a shut immersion $C\to X$ over $S$? Simply to be clear: A projective contour is a smooth projective morphism $X\to S$ such that the geometric fibers are geometrically attached and also of measurement 1. In strai...
2022-07-17 15:17:11

Is the Fermat scheme $x^p+y^p=z^p$ always normal

Let $K$ be a number field with ring of integers $O_K$. Is the closed subscheme $X$ of $\mathbf{P}^2_{O_K}$ given by the homogeneous equation $x^p+y^p=z^p$ normal? I know that this is true if $K=\mathbf{Q}$. (My method of proof is a bit awkward.) I expect the answer to be no (unfortunately) in general. What is an equation for the normalization of...
2022-07-16 17:18:22

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ over $C$, i.e., a fibered surface $\mathcal{X}\to C$. Are there any other applications or problems where one considers a model for $X$ over $C$ to ...
2022-07-12 15:04:08

What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). Example. Take $\Gamma = \Gamma(n)$. Then $X_\Gamma = X(n)$. What are the branch points of $X_\Gamma \to X(1)$. Are they just the three points given...
2022-07-12 14:38:30

Is the degree of a Galois morphism bounded by $84(g-1)$

Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$. Assume $g=g(X) \geq 2$. Is the degree of $X\to Y$ bounded by $84(g-1)$? I think the answer is yes. In fact, the degree of $X\to Y$ is the cardinality of $\# \mathrm{Aut}(X/Y)$. This is bounded from above by $\# \mathrm{Aut}(...
2022-07-12 14:27:02

Does the absolute Galois group act on the moduli space of curves

Do elements of the absolute Galois group $G=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ of $\mathbf{Q}$ induce automorphisms of $\mathbf{C}$ if we extend the morphisms trivially to the rest of $\mathbf{C}$? Here we consider an element of $G$ as an automorphism of $\overline{\mathbf{Q}}\subset \mathbf{C}$ (and we fix this embedding). Does thi...
2022-07-11 23:45:34

For curves, is being defined over a number field invariant under birational equivalence

Suppose that a (attached) Riemann surface area $X$ is birational to a Riemann surface area $Y$ which can be specified (algebraically) over the area of algebraic numbers. Does this indicate that $X$ itself can be specified over the area of algebraic numbers? Primarily, I'm asking if the building "can be specified over the area of algebraic...
2022-07-11 23:45:28

Does there exist a finite morphism of algebraic curves such that...

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that $Y$ can not be defined over $K$. Is it possible that $X$ can still be defined over $K$? Equivalently, suppose that $X$ can be defined over $K$. Then is...
2022-07-11 23:27:16

Are these two notions of Galois morphism the same

Let $f:X\to Y$ be a limited morphism of indispensable systems. Allow $G$ be the automorphism team of $X$ over $Y$. Are the adhering to 2 problems equal? The function area expansion $K(Y)\subset K(X)$ is Galois (in the area - logical feeling) The quotient $X/G$ exists and also the all-natural morphism $X/G\to Y$ is an isomorphism.
2022-07-11 06:23:56

Curve - non singular curve and its genus

Help me please with this trouble: $ X \subset \mathbb{P}^{2}$ specified as $x^{3}y+y^{3}z+z^{3}x=0$ 1. Confirm X - non single contour and also locate its category. 2. Confirm X - topmost contour over $F_{8}$ area, and also locate all factors of X. Thanks a whole lot!
2022-07-10 07:51:16

Is a cover Galois if and only if it is geometrically Galois

Let $K$ be a number area and also allow $\pi:X\to \mathbf{P}^1_K$ be a limited morphism, where $X$ is a smooth projective geometrically attached contour. Is $\pi$ a Galois cover if and also just if the base adjustment $\pi_{\overline K} : X_{\overline K} \to \mathbf{P}^1_{\overline K}$ is a Galois cover?
2022-07-10 05:00:46

Does composing the Frobenius with an automorphism give another Frobenius

Let $X_0$ be a selection over $\mathbf{F}_q$. Take into consideration the Frobenius $F_0:X_0\to X_0$. Allow $X= X_0\times \bar{\mathbf{F}_q}$ and also allow $F:X\to X$ be $F_0 \times \textrm{id}$. Allow $f:X\to X$ be an automorphism of limited order. Is $F\circ f$ the Frobenius relative to some new means of decreasing the base area to $\mathbf{...
2022-07-07 02:33:24

Why is the trace map on an abelian variety continuous

Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$. Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i \mathrm{Tr}(t_a^\ast, H^i(X,\mathbf{C}))$$ continuous? Here I consider the usual singular cohomology with $\mathbf{C}$-coefficients. (The coefficients d...
2022-07-06 23:53:44

how does one intersect the diagonal with a graph on the surface $X\times X$

I intend to do a concrete instance of a junction item for myself. Take into consideration the endomorphism $f:\mathbf{P}^1_k\to \mathbf{P}^1_k$ offered by $(x:y)\to (y:x)$. It has specifically 2 set factors: $(1:1)$ and also $(1:-1)$. I intend to calculate that the junction item on $\mathbf{P}^1\times \mathbf{P}^1$ is 2. I assume I can in some...
2022-07-06 23:30:32