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Let $K$ be a number field, $S$ be a finite set of places of $K$, and $K(S,p)$ be the $p$-Selmer group of the $S$-integers of $K$, that is the set of nonzero elements of $K$ modulo $p$-th powers whose valuation at $\mathfrak{p} \notin S$ is $0$ modulo $p$.

Let $Z$ be the subgroup of $K(S,p)$ of elements whose norm is a $p$-th power. It comes up in descent calculations on elliptic curves.

Then $Z$ is an $\mathbb F_p$-vector space, of finite dimension say $d$. In particular it has a basis. Since $Z$ is multiplicative, its elements are considered as powers of the basis elements, the powers considered mod $p$.

Suppose though that we do not know a basis, rather we have $2d$ random elements of $Z$, but we do know they are a spanning set for $Z$, not linearly independent.

If I am given an arbitrary element $z_1$ of $Z$ as an ordinary looking element of $K$, is it possible to express $z_1$ efficiently as a product of powers of the spanning set elements?

This question is highly interesting to me. There is an established way to find an actual basis for $Z$ by computing generators for the class and unit groups of $K$, but this is computationally expensive (e.g. see [1]).

So I am wondering is there a way to get around that difficult computation, provided we can efficiently generate a spanning set. It would be a powerful computational tool to be able to write arbitrary elements of $Z$ in terms of the spanning set.

The most straightforward way for me to write down this question was to just state the exact context I'm interested in, rather that try to make it more abstract/as general as possible. But there is probably a very simple question behind it about spanning sets for a vector space, not having a basis, writing elements in terms of the spanning set elements. I couldn't quickly find an answer to this question. If there is a very obvious answer please tell me and I will accept it.

[1] Djabri, Z.; Schaefer, Edward F.; Smart, N. P. Computing the $p$-Selmer group of an elliptic curve. Trans. Amer. Math. Soc. 352 (2000), no. 12, 5583--5597

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  • $\begingroup$ The valuation maps to $\mathbb{F}_p^{n}$, where $n$ is the number of finite places in $S$, should reduce the problem to the case $S=\varnothing$. And then one can reduce modulo small primes of norm congruent to $1$ modulo $p$, though the discrete logarithm there could be harder to work with. $\endgroup$
    – Chris Wuthrich
    Commented 1 hour ago
  • $\begingroup$ That simplifies the problem slightly, there remains writing $z_1 \in Z$ in terms of a spanning set. I believe I have a way of producing such a spanning set efficiently. It would be no use though unless the main problem is somehow easy. $\endgroup$
    – Oisín Robinson
    Commented 1 hour ago

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