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Let $G$ be a simply connected semi-simple linear algebraic group over a global field $k$. The Hasse principle for algebraic groups states that the map $$H^1(k,G)\rightarrow\prod_vH^1(k_v,G)$$ is injec …

Here's an argument that one should expect that the two numbers gotten this way must be transcendental. Really, what I am showing is that the locus of $(a,b)$ in $\mathbb{R}\times\mathbb{R}$ where one …

Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot o …

Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid cohomo …

An update from 2018: There has been some recent work on a heuristic suggesting that there are infinitely many elliptic curves of every rank $<21$ but only finitely many of rank $> 21$ (it's unclear to …

As discussed in the comments, I don't see how to extract the desired matrix from the original question (about spanning the vector space). However, the matrix being nonzero IS equivalent to the followi …

Note that the defining equation for $X$ can be rewritten as $$(x+y)^3+(x-y)^3+(z+w)^3+(z-w)^3=-2.$$ As the linear transformation $(x,y,z,w)\mapsto(x+y,x-y,z+w,z-w)$ is invertible over any field of cha …