Computing Selmer groups associated to mod p Galois representations

We present methods to compute Selmer groups associated to mod p Galois representations rho over a number field K, with a particular focus on comparing their ranks with periods coming from cohomology classes associated to rho by Serre’s conjecture. This provides evidence for a loose version of a “mod p Bloch-Kato conjecture”, where the vanishing of a period is predicted to capture the presence of rank in a Selmer group. Our methods are explicit, and implemented in Magma.

💡 Research Summary

The paper develops explicit algorithms for computing Selmer groups attached to mod p Galois representations ρ over a number field K, and uses these computations to test a “mod p Bloch–Kato” type relationship between Selmer ranks and periods coming from cohomology classes predicted by Serre’s conjecture.

The authors begin by fixing a continuous two‑dimensional representation ρ : G_K → GL₂(𝔽_q) with p the characteristic of 𝔽_q. Because the image of ρ is finite, it factors through a finite Galois extension L/K. For each finite place p of K they introduce the decomposition group D_p, the inertia group I_p, and, when ρ is nearly‑ordinary at p, a one‑dimensional D_p‑stable subspace ℓ ⊂ V. Three families of local conditions are defined: (i) relaxed (no restriction), (ii) unramified (kernel of the restriction to inertia), and (iii) nearly‑ordinary (kernel of the restriction to inertia after modding out by ℓ). These give rise to three Selmer systems L_rel, L_NO, L_unr and the corresponding Selmer groups Sel_rel(ρ), Sel_NO(ρ), Sel_unr(ρ). By construction Sel_unr ⊆ Sel_NO ⊆ Sel_rel.

A key technical step is the use of the inflation‑restriction exact sequence. The authors show that, under the usual vanishing of H¹(Gal(L/K),V) and H²(Gal(L/K),V) (which holds for all groups they consider), the global cohomology group H¹(G_K,V) is canonically isomorphic to the space of Gal(L/K)‑equivariant homomorphisms Hom_{Gal(L/K)}(G_L,V). Each non‑zero 𝔽_q‑line ⟨f⟩ in this space determines a Galois extension M_f/L defined by M_f = L^{ker f}. Theorem 4.3 proves that M_f/L is abelian with Galois group V, that M_f/K is Galois, and that the action of Gal(L/K) on Gal(M_f/L) is exactly the representation ρ. Conversely, any such extension yields a line in Hom_{Gal(L/K)}(G_L,V).

The computational pipeline, implemented in Magma and released on GitHub, proceeds as follows: (1) given ρ, compute the field L and the groups D_p, I_p; (2) compute the space Hom_{Gal(L/K)}(G_L,V) via cohomology; (3) extract the subspaces satisfying each local condition; (4) count the number of independent lines, which equals the 𝔽_p‑dimension (rank) of the corresponding Selmer group; (5) for each line construct M_f and verify the local conditions using class‑field‑theory routines.

The authors apply the method primarily to the p‑torsion representations of elliptic curves obtained from the LMFDB. For each curve they compute the three Selmer ranks and compare them with the “period” attached to the Serre‑type cohomology class. Empirically they observe a clear pattern: the nearly‑ordinary Selmer rank is zero exactly when the period is non‑zero, and positive when the period vanishes. Moreover, the difference between the nearly‑ordinary rank and the (conjectural) true mod p Bloch–Kato rank never exceeds one, mirroring the well‑known over‑estimation property of the Greenberg Selmer group in the p‑adic setting.

Technical subtleties are addressed. The authors note that H²(Gal(L/K),V) can be non‑trivial for certain non‑abelian images (e.g., S₃ in GL₂(𝔽₃)), but in all their examples the cohomology groups vanish, so the isomorphism H¹(G_K,V) ≅ Hom_{Gal(L/K)}(G_L,V) holds. They also prove that the Selmer ranks are independent of the choice of the place q of L lying above a given p, by showing that the decomposition and inertia groups at different q’s are conjugate inside Gal(L/K).

In conclusion, the paper provides a concrete, reproducible method for computing mod p Selmer groups and supplies experimental evidence for a weak mod p Bloch–Kato conjecture: the vanishing of a Serre‑type period predicts the existence of non‑trivial Selmer classes, with at most a one‑dimensional discrepancy. The work opens several avenues for future research, including higher‑dimensional representations, non‑nearly‑ordinary cases, and a deeper connection to p‑adic L‑functions and Iwasawa theory.