galois modules
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2018 ◽  
Vol 239 ◽  
pp. 294-321
Author(s):  
DAVID BURNS

We investigate the Galois structures of $p$-adic cohomology groups of general $p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded $p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.


2017 ◽  
Vol 165 (1) ◽  
pp. 137-162 ◽  
Author(s):  
ZEXIANG CHEN

AbstractWe compute certain twists of the classical modular curve X(8). Searching for rational points on these twists enables us to find non-trivial pairs non-isogenous elliptic curves over ℚ whose 8-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over ℚ.


2016 ◽  
Vol 46 (5) ◽  
pp. 1405-1446 ◽  
Author(s):  
Sunil Chebolu ◽  
Ján Mináč ◽  
Andrew Schultz
Keyword(s):  

2015 ◽  
Vol 2016 (11) ◽  
pp. 3205-3236
Author(s):  
Erik Jarl Pickett ◽  
Lara Thomas
Keyword(s):  

2014 ◽  
Vol 10 (01) ◽  
pp. 1-12
Author(s):  
ALEX BARTEL

We compare two approaches to the study of Galois module structures: on the one hand, factor equivalence, a technique that has been used by Fröhlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand, regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We then use this comparison to derive a factorizability result on higher K-groups of rings of integers, which is a direct analogue of a theorem of de Smit on S-units.


2014 ◽  
Vol 17 (1) ◽  
pp. 536-564 ◽  
Author(s):  
Tom Fisher

AbstractWe use an invariant-theoretic method to compute certain twists of the modular curves $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X(n)$ for $n=7,11$. Searching for rational points on these twists enables us to find non-trivial pairs of $n$-congruent elliptic curves over ${\mathbb{Q}}$, that is, pairs of non-isogenous elliptic curves over ${\mathbb{Q}}$ whose $n$-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over ${\mathbb{Q}}(T)$, and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves  over ${\mathbb{Q}}$.Supplementary materials are available with this article.


2013 ◽  
Vol 88 (3) ◽  
pp. 845-859 ◽  
Author(s):  
Alex Bartel ◽  
Bart de Smit
Keyword(s):  

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