Bounds on $p$-primary Selmer groups when $E[p]$ is irreducible

Question

My question is: if $E$ is an elliptic curve over $\mathbf{Q}$, and $p$ is a prime number such that $E[p]$ is irreducible as a Galois module, how does one go about bounding the $p$-primary Selmer group of $E$? Is this known to be a difficult / intractable problem, or are there known techniques to attack it?

If $E[p]$ is reducible, I've seen a few examples of how you can bound the $p$-primary Selmer group from above. For example, these notes by Ralph Greenberg show that the curve $E = 11a1$ has $5$-primary part of Selmer equal to $0$ (see page 74, second paragraph). The proof heavily relies on the fact that $E[5]$ is reducible, and the same proof doesn't go through for primes $p$ where $E[p]$ is irreducible.

Are there any known techniques to bound Selmer groups in the case that $E[p]$ is irreducible? Or perhaps easier, are there techniques available to prove algebraically (i.e: without BSD) that the $p$-primary Selmer group of $11a1$ is zero for primes $p \neq 5$? Any references would be appreciated.

Answer 1

The question is a bit too broad. I will make a few comments that may answer some of the questions that seem to be behind it.

For a given, fixed elliptic curve $E$ over a number field $K$, the Selmer group for $E[p]$ is in principle computable. However in practice this is very difficult and not really feasible for a prime $p>10$ especially if $E[p]$ is irreducible. There is a vast amount of literature about the computational methods to determine the Selmer group and quite a few implementations ($p$-descents). When $E[p]$ is reducible, the problem is easier as one can compare it to the Selmer groups of the isogenies that appear. In practice one can always extend the field $K$ to a field where $E[p]$ becomes reducible or even trivial as a Galois module. Then the Selmer group becomes easier to calculate, but the class group and the units that are involved will be harder to determine. The last step is to restrict to $K$. However, this can all be done over $K$ in general when considering the Selmer group via étale algebras. Reference in and to the papers 34,38, and 46 in this list is a good place to learn about this.

Not sure what "algebraically" means. But even for a fixed elliptic curve over $\mathbb{Q}$, the methods to calculate the Selmer groups for all $p$ only work for curves of analytic rank $0$ or $1$ and in all cases they use the fact that the curve is modular or that it has complex multiplication. It is the presence of an Euler system that will allow one to bound the Selmer groups in these cases in terms of the leading term of the complex $L$-function. In practice, this is proving BSD. For curves of analytic rank $2$ or larger, one can use the known results from Iwasawa theory to calculate the Selmer group for a fixed $p$ by linking it to the leading term of the $p$-adic $L$-function. But again this uses modular symbols - though their calculation is sort of "algebraic".

Finally, there are many results on bounding the Selmer group in families of elliptic curves, like the work of Bhargava and his co-authors, or questions about how the Selmer group varies in families of quadratic twists. But again, this is too vast an area to answer this here. In some sense, most articles out there containing the word "Selmer group" in the title will be interested in bounding it.

Answer 2

Your question "how does one go about bounding the $p$-primary Selmer group of $E$" is not precise, because you don't specify what sort of bound you want. Building onto one piece of Chris Wuthrich's nice answer, it's not hard to get a bound that depends on (for example) the prime $p$, the number of primes of bad reduction, and the $p$-rank of the ideal class group of the field $\mathbb Q(E[p])$. (One can do better.) Then one can use a bound for the size of the entire class group of $\mathbb Q(E[p])$ (which depends on the discriminant and degree of the field, but those only depend on $p$ and the primes of bad reduction). So it is possible to derive a completely effective upper bound $$ \text{rank of $p$-Selmer group of $E$} \le F(p,S_E), $$ where $S_E$ is the set of primes of bad reduction for $E$. Of course, this upper bound is likely to be far larger than the truth.

There is a completely explicit bound for the rank of the Mordell-Weil group of an abelian variety $A/K$ in a short note by Oee and Top (Takeshi Ooe and Jaap Top, On the Mordell-Weil rank of an abelian variety over a number field, J. Pure Appl. Algebra 58 (1989), no. 3, 261–265). To make it completely explicit, they use $A[2]$, but if you work through the proof, you'll see it gives something similar using $A[m]$ for any $m$. See in particular their Theorem 2, which gives a bound for the rank of $A(K)/mA(K)$ in terms of the number of primes of bad reduction and the rank of the $m$-torsion subgroup of the ideal class group of $K(A[m])$. And if you check the proof, you'll see that that they are actually bounding the rank of the Selmer group $S^{(m)}(A/K)$, and then they use that as an upper bound (more or less) for the rank of $A(K)$.