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](https://arxiv.org/abs/math/9809206) 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.



Bounds on $p$-primary Selmer groups when $E[p]$ is irreducible
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](https://homepages.warwick.ac.uk/staff/J.E.Cremona/papers/) 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.


