Daniel Loughran
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 Nov 28 comment Arithmetically equivalent number fields and Langlands Program The isomorphism from Prasad's paper comes from the fact that there is isomorphism of algebraic groups $\mathrm{R}_{F/\mathbb{Q}} \mathbb{G}_m \cong \mathrm{R}_{F'/\mathbb{Q}} \mathbb{G}_m$ over $\mathbb{Q}$ (here $\mathrm{R}_{F/\mathbb{Q}}$ denotes the Weil restriction). Perhaps you could obtain an isomorphism of the associated general linear groups is a similar way...? Nov 28 comment Arithmetically equivalent number fields and Langlands Program Perhaps you could edit your question to try to make it clearer? Nov 28 comment Arithmetically equivalent number fields and Langlands Program I agree with eric that the quantifiers in you question are not clear. Anyway, a theorem of Neukirch-Uchida states that any number field is determined by its absolute Galois group: en.wikipedia.org/wiki/Neukirch%E2%80%93Uchida_theorem. This suggests to me that all the automorphic data associated to a number field should determine it (though of course I can't make this more precise). Nov 22 comment Rational points on the “quintic circle” $x^5 + y^5 = 7$ The modular techniques and Frey curve machinery used to prove FLT could also possibly be made to apply in your case, but I am not an expert in these methods. Nov 22 comment Rational points on the “quintic circle” $x^5 + y^5 = 7$ It helps to instead consider the corresponding projective curve $X^5 + Y^5 = 7Z^5$ in $\mathbb{P}^2$ of genus $6$. This has a rational point, namely $(X:Y:Z) = (1:-1:0)$. Your conjecture is that this is the only rational point. The similarity here to the $n=5$ case of Fermat's last theorem should be very clear. Of course the proof of FLT for all $n$ was very hard, however there exist elementary proofs for the $n=5$ case (en.wikipedia.org/wiki/…). Perhaps these could be adapted to your case. Nov 20 revised Automorphisms and infinitesimal deformations of a smooth complete intersection added 9 characters in body Nov 19 comment Arithmetic zeta function and local zeta functions @R. van Dobben de Bruyn: This is the standard definition of an arithmetic zeta function: en.wikipedia.org/wiki/Arithmetic_zeta_function. No good reduction hypotheses are required. The OP should really however be choosing a model for $X$ over $\mathbb{Z}$. Nov 16 answered Automorphisms and infinitesimal deformations of a smooth complete intersection Nov 16 answered The number of integral solutions to $x^2+y^2-az^2=0$ Nov 16 comment Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? I don't think this question is so far fetched. Granville has shown that a suitably generalised version of the ABC conjecture would imply the non-existence of Siegel zeros for quadratic Dirichlet L-functions. Nov 12 answered Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension? Nov 11 comment Intuition for Zagier's theorem for $\zeta_K(2)$ Perhaps this follows from Tamagawa number considerations, as in mathoverflow.net/questions/111339/…? Nov 9 answered How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random? Nov 9 comment Galois cohomologies of an elliptic curve Does one not actually have $H^2(\text{Gal}(\bar K/K),\mathsf{A}_{\bar K}^*)=\prod_{v \text{ real}} \mathbb Z/2\mathbb Z \prod_{ v\nmid \infty} \mathbb Q/\mathbb Z$, where products are over places $v$ of $K$? Nov 8 comment Galois cohomologies of an elliptic curve I think you need to be slightly more careful with the real places. Namely, the cohomology $H^i(\mathbb{R}, E[m])$ can be non-zero for $i > 2$; though of course it is periodic in $i$ with period $2$. Also, for number fields with real embeddings one can have a non-trivial $H^3$. Nov 4 comment Explicit $2$-descent on elliptic curves This is exactly what I was after, thanks Michael! Nov 4 accepted Explicit $2$-descent on elliptic curves Nov 3 comment Explicit $2$-descent on elliptic curves Thanks for clarifying. Yes you are right I did not make myself clear enough. What I meant to say was ELS => $k$-rational divisor degree $2$, and we don't always have such a divisor in general. Nov 3 comment Explicit $2$-descent on elliptic curves Thanks for the reference, but I'm not quite sure this achieves what I want. If I understand it correctly, over a number field $k$, it is only the everywhere locally soluble homogeneous spaces which can be written as $y^2 = \mbox{quartic}$ (being everywhere locally soluble implies that there is an actual $k$-rational divisor of degree $2$). Cremona only seems to handle such spaces in his notes. However, I want equations for all such homogeneous spaces, not necessarily just those with a $k$-rational divisor of degree $2$. Nov 3 asked Explicit $2$-descent on elliptic curves