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Let $k$ be a local field with absolute Galois group $\Gamma_k$, inertia subgroup $I_k \subset \Gamma_k$, and residue field $\mathbb{F}$. Let $M$ be a finite Galois module over $k$. The unramified Galo …

What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given. Note that …

If the Neron-Severi group of an abelian variety is $\mathbb{Z}$, then a principal polarisation, if it exists, is unique. This is the generic case, as well as for generic jacobians. To find counter-exa …

The "best" way to deal with quadratic forms using the circle method is via Heath-Brown's delta symbol method. You can read about this in detail in the paper: Heath-Brown - A New Form of the Circle Met …

Shparlinski has done a lot of work on problems like this, in the more general setting of linear recurrence sequences. (Your sequence of Mersenne numbers is such a sequence). I'd recommend looking at C …

This nice approach to points on schemes in fact becomes crucial once one leaves the world of schemes and travels to the galaxy of stacks. For an algebraic stack $X$, one defines a point of $X$ to be a …

See Section 5.7 of Serre's Galois Cohomology. (In general Chapter 5 of this book is a fairly definitive reference for non-abelian cohomology, and he works with an arbitrary profinite group $G$).

Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes. Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then …

It is difficult to get far in the modern theory without some algebraic geometry. This is the approach taken in the book: Bjorn Poonen, Rational points on varieties, Graduate Studies in Mathematics 18 …

Serre deals with problems of this type in the paper: Serre -Divisibilité de certaines fonctions arithmétiques The fact you want should follow from the results in Sections 1 and 2. Alternatively, th …

Yes this is the fact that for a non-trivial irreducible Artin character, the associated Artin L-function is holomorphic and non-zero on $\rm{re}\, s \geq 1$. For the trivial character, one just obtai …

The answer to the precise question is yes. See Theorem 1.2 of: "Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields," https://arxiv.org/pdf/0912.0325.pdf. …

This is false. Take $X$ to be a smooth plane conic without a rational point. Consider the surface $$S = X \times X.$$ Over the algebraic closure this becomes isomorphic to $\mathbb{P}^1 \times \mathbb …

As explained in the comments, there is no such variety. This is an application of a general set of techniques called "spreading out". You can find a very nice treatment of this in Chapter 3 of the bo …

Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$. For a paper, I need the result that $$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\v …