1 · The essentials
These manuscripts study objects describing certain symmetries of numbers, using arithmetic based on the prime 2. The main result claims that every object of a specified kind fits into a large structure built from modular forms. The objects must, in particular, be two-dimensional, continuous, odd, and absolutely irreducible, and have ramification at only finitely many primes. The result includes cases where their simplified versions modulo 2 become scalar or reducible. Belonging to this completed structure does not, by itself, mean that they come from classical modular forms. A companion manuscript claims that classical connection, up to a transformation called a Tate twist, under additional conditions at the prime 2. Another describes the dimension of the modular structures involved. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
An imperfect image is a library: the main result says that every file meeting the criteria can be accommodated in a vast catalogue, possibly by expanding the section where it belongs. This does not yet guarantee that it corresponds to a classical book; the third manuscript claims that correspondence under additional conditions.
Why it matters
The significance is in treating the prime 2 without excluding scalar or reducible residual representations. The manuscripts distinguish two goals: occurring in a completed Hecke algebra and being classically modular. The third presents its result as resolving the odd, regular, two-dimensional Fontaine–Mazur conjecture over at the prime 2.
2 · Going further
The main problem concerns a representation of , the absolute Galois group of . It is assumed to be continuous, odd, absolutely irreducible, two-dimensional and 2-adic, and unramified outside finitely many finite primes. The first manuscript claims that it occurs in the full completed Hecke algebra at some odd tame level. That level may contain auxiliary tame primes; scalar and reducible residual representations are included.
This is a pro-modularity result, not an assertion of classical modularity. No de Rham hypothesis is imposed in this first statement. The supplied abstracts do not describe the steps of the argument: they identify the goal—realizing the representation in a completed Hecke algebra—but do not provide a method of demonstration to explain.
The second manuscript claims that, for every odd positive integer , each irreducible component of the full two-adic Hecke algebra of level has Krull dimension four. It presents this as the case of Emerton’s dimension conjecture, including all residual components. The third claims modularity up to Tate twist for continuous, irreducible, odd, two-dimensional representations unramified outside finitely many primes that are de Rham at 2 with distinct Hodge–Tate weights. The three texts thus address completed occurrence, dimension, and classical modularity; the abstracts do not specify their logical dependencies.
Useful notions
- Groups and two-dimensional representations
- The absolute Galois group of
- 2-adic numbers and residual reduction
- Modular forms and Hecke algebras
- De Rham conditions and Hodge–Tate weights
What was known before / what is new
Before
The source places these works in the context of Emerton’s dimension conjecture and the Fontaine–Mazur conjecture. It supplies neither a history of earlier results nor a precise account of previously established cases.
New
The first manuscript claims occurrence in a completed Hecke algebra without a de Rham hypothesis, including scalar and reducible residual cases. The second claims dimension four for every irreducible component at odd levels. The third claims classical modularity up to Tate twist under the de Rham and distinct-weight hypotheses, without residual restrictions.
Limits and assumptions
“Unrestricted” does not remove the hypotheses: the main result concerns only continuous, odd, absolutely irreducible, two-dimensional 2-adic representations unramified outside finitely many primes. The resulting odd level may include auxiliary primes. This result does not assert classical modularity; the third manuscript asserts it up to Tate twist with additional hypotheses at 2. The main result is not formalized in Lean and has not been formally verified. The abstracts do not supply the arguments or list the questions remaining open beyond these statements.
Mini glossary
- 2-adic representation —
- A way of describing a group action by matrices whose coefficients belong to 2-adic arithmetic.
- Residual representation —
- The version obtained by reducing a 2-adic representation to characteristic 2.
- Absolutely irreducible —
- A representation with no nonzero proper invariant subspace, even after extending its coefficient field.
- Completed Hecke algebra —
- A completed algebraic structure collecting Hecke operators associated with modular forms.
- Krull dimension —
- A measure of a ring’s dimension based on chains of strict inclusions among its prime ideals.
- Tate twist —
- A modification of a representation by a power of the cyclotomic character.
Keywords : Pro-modularity · 2-adic Hecke algebras · Fontaine–Mazur
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Unrestricted pro-modularity at the prime two
06/10/2026Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.
The Dimension of the Two-Adic Hecke Algebra at Odd Level
05/10/2026For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.
Fontaine–Mazur modularity at the prime 2
06/10/2026We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of , unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.