6

For (3), the answer is no, because there is no irreducible element in $\overline{\mathbb{Z}}$: for example, every element is a square. Similarly for (1), the answer is no because there are infinite ascending chains of ideals: for example, $(x) \subset (x^{1/2}) \subset (x^{1/4}) \subset \cdots$, where $x$ is not a unit (e.g. $x=2$) and $\{x^{1/2^n}\}$ is a ...


5

I think we can describe $P$ a bit more, using Dirichlet’s unit theorem. Since $\overline{\mathbf{Z}} = \varinjlim_{[K:\mathbf{Q}] < \infty} \mathcal{O}_K$, the same is true for the units. Now Dirichlet tells us that taking logs of the archimedean valuations $v \mid \infty$ of $K$ gives us an embedding: $$\mathcal{O}_K^\times/\mu_\infty(K) \hookrightarrow \...


4

If you could solve this problem in polynomial time, then NP would be contained in BPP, which is viewed as being approximately as unlikely as P = NP. Too see this, pick your favorite encoding of SAT into diophantine equations on $\{0,1\}^n$ (for instance, you can take $f$ to be a sum of squares of expressions corresponding to individual clauses), and apply ...


4

Let $E/\mathbb{Q}$ be an elliptic curve. There exist positive integers $d_p$ and $e_p$, with $d_p|e_p$, such that group $E(\mathbb{F}_p)$ is isomorphic to $\mathbb{Z}/d_p\mathbb{Z} \times \mathbb{Z}/e_p\mathbb{Z}$. Kowalski conjectured that there exists a constant $c_E>0$ such that $\sum_{p\leq x}d_p\sim c_E f_E(x)$, where $f_E(x)=x$ if $E$ has CM and $...


4

The ring $\overline{\mathbb{Z}}$ is called the ring of algebraic integers. You can find information about prime ideals, e.g., in https://math.stackexchange.com/questions/156231/non-zero-prime-ideals-in-the-ring-of-all-algebraic-integers.


4

It is always injective, and the cokernel is the group of Brauer classes split by $k(C)$, see On the period-index probem in light of the section conjecture by J. Stix. Equivalently, those are the classes of Brauer-Severi varieties $P$ with a morphism $C\to P$. If $k$ is a number field and $C$ has local points everywhere, it cannot map to a non-trivial Brauer-...


3

The set $J(C)_{\Theta}[n]$ has the structure of a smooth irreducible algebraic curve, and the restriction of $J(C)\xrightarrow{\times n } J(C)$ to $C$ defines a morphism $J(C)_{\Theta}[n]\rightarrow C$ which is a finite etale cover with Galois group $J(C)[n]$. I don't think it is good to think of $J(C)_{\Theta}[n]$ as generalized torsion points. I think it's ...


3

(1) No, as Brunault explaned, we may get an infinite series of ideals. In more general setting, if $A$ is an integral domain then any infinitely ramified (Indeed, an integral extension $B$ of $A$ is "infinitely ramified" if there is a tower $A\subsetneq A_1\subsetneq A_2\subsetneq \cdots$ such that $A_{i+1}/A_i$ are ramified) integral extension of $...


2

A partial (I have a few questions at the end) proof to Q.2,3 communicated to me by Ananth Shankar: Suppose $\mathscr L$ is a line bundle corresponding to a Galois stable element in $Pic_C(\overline{k})$ and assume also that it is very ample. Let $f: C \to \mathbb P(H^0(C))$ be the induced map on global sections defined over $\overline{k}$. This map is in ...


2

I think Kisin's article "Crystalline representations and $F$-crystals" may be helpful. If you haven't read this, the following is a quick mention of some results. There are several ways to express $T_pG$ in the form of a $\mathbf{Z}_p[G_{K_{\infty}}]$-module. For example, using Breuil-Kisin modules (cf. Corollary 2.1.4; Lemma 2.2.4; Theorem 2.2.7). ...


2

The corrected formulation of the K"unneth formula in the Stacks project can also be used to given an answer this question. Let me explain. Observe that the differentials in the de Rham complex $E \otimes \Omega^\bullet_{X/S}$ of $(E, \nabla_E)$ are differential operators of finite order and similarly for $(F, \nabla_F)$. The construction in Section 0G4A ...


1

Step 1 indeed fails. The functor from BKF-modules over $C^+$ to shtukas over $C^+$ is not going to be full. You can still glue so you still get an essentially surjective functor but there could be more interesting $\varphi$-modules over $\mathcal{Y}_{(0,\infty]}$ by which you could attempt to glue. If $k=O_C/\mathfrak{m}$ and $k^+=C^+/\mathfrak{m}$ then the ...


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