Galois representations — flashcards
Every card the map can derive about Galois representations — definitions, statements, attribution, and the edges it sits on.
Theorems 5
State: The modularity theorem.
Every elliptic curve over ℚ is modular. Proved for semistable curves by Wiles with Taylor (1995), in general by Breuil, Conrad, Diamond and Taylor (2001). Fermat's Last Theorem is a corollary.
derived from st/modularity by the statement rule
State: The geometric Langlands conjecture.
An equivalence between categories of local systems on a curve and sheaves on the moduli of bundles over it. Proved 2024 by Gaitsgory, Raskin and collaborators.
derived from st/geometric-langlands by the statement rule
State: The strongest non-numerical evidence for the zeta–RMT correspondence is a proven theorem that explicitly does not transfer.
Katz–Sarnak's theorem for curves over finite fields shows Frobenius eigenvalues — a genuine, proven arithmetic object — converge to classical compact-group statistics, the strongest available non-numerical evidence for the zeta–random-matrix correspondence. Its scope is explicitly bounded: there is no known analogue of Frobenius or étale cohomology over ℤ, so the theorem cannot be strengthened into evidence for the Riemann Hypothesis itself, only for the shape the correspondence takes in a genuinely provable adjacent setting. The logical endpoint of pushing the whole analogy is the Hilbert–Pólya conjecture: the search for a self-adjoint operator whose spectrum is literally the zeros. The CFKRS conjecture — an exact splitting of zeta's moment-asymptotics into a universal random-matrix factor times an arithmetic Euler-product factor — is the structural tool proposed for locating the correspondence's boundary without further numerical computation, though the arithmetic factor itself was named in the source conversation, not computed or explored there.
derived from st/katz-sarnak-hilbert-polya-boundary by the statement rule
State: A constructed instance: Frobenius eigenvalues of an actual curve family, verified against Weil's theorem and tracked toward symplectic statistics.
st/dwork-rationality establishes the zeta function of a variety over a finite field is rational (the first Weil conjecture); the second Weil conjecture (the 'Riemann Hypothesis for curves') states its zeros satisfy |α|=√q. Both were made concrete rather than merely cited: a genus-2 hyperelliptic curve over F_11 was built, its point counts over F_11 and F_121 fed through Newton's identities (st/newton-identities) to obtain the Frobenius characteristic polynomial exactly, and its roots verified to satisfy |α|=√q to eight decimal places. This was then scaled into an actual statistical family — 101 curves at q=11, then 861 at q=31 using a vectorized finite-field-arithmetic implementation — pooling Frobenius angles and tracking the emergence of symplectic-ensemble (USp(4)) endpoint repulsion as q grows, the predicted structure for this family type.
derived from st/frobenius-curve-family-weil-verification by the statement rule
State: Perfectoid spaces.
A tilting equivalence relating characteristic 0 and characteristic p geometry, which turned a long list of separate p-adic comparison results into instances of one construction.
derived from st/perfectoid by the statement rule
Names and dates 3
The modularity theorem — who, and when?
Wiles with Taylor; then Breuil, Conrad, Diamond and Taylor, 1995, completed 2001.
derived from st/modularity by the attribution rule
The geometric Langlands conjecture — who, and when?
Gaitsgory, Raskin and collaborators, 2024.
derived from st/geometric-langlands by the attribution rule
Perfectoid spaces — who, and when?
Peter Scholze, 2012.
derived from st/perfectoid by the attribution rule
Connections 2
What is the correspondence between Galois representations and Modular & automorphic forms? [Correspondences]
Galois representations ↔ automorphic representations, matched by their L-functions. Proved for elliptic curves over ℚ (modularity, Wiles–Taylor, giving Fermat) and in many other cases; open in general.
Recorded as conjectural: expected, not proved.
derived from correspondences:galois-representations→modular-forms by the connection rule
What is the correspondence between Galois representations and L-functions? [Correspondences]
Each Galois representation produces an L-function — the arithmetic side's contribution to the shared currency.
derived from correspondences:galois-representations→l-functions by the connection rule
Concepts 1
What is Galois representations, in one sentence?
Actions of a field's absolute Galois group on vector spaces. The arithmetic side of the Langlands correspondence.
derived from galois-representations by the concept rule