The étale fundamental group of the Néron model of an abelian variety over K is the semidirect product of a finite group with the étale fundamental group of O_K.
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Ring objects in the equivariant de- rived Satake category arising from Coulomb branches
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For primes N and p with N ≡ 1 mod p, the rank r of Mazur's Eisenstein Hecke algebra equals one plus the vanishing order of a mod-p zeta element interpolating L-values at -1 when r is 2 or 3, with a uniform extension to level N² and partial results for higher ranks.
The coordinate ring of the universal centralizer equals the result of applying Demazure operators to the coordinate ring of X precisely when the W-fixed points of the Weil restriction of X is an integral scheme.
Generalizes Steinberg's centralizer component description to unipotent elements under non-etale covers, with applications to L-parameter moduli multiplicities, deformation rings, and non-existence of Springer isomorphism for PGL_p in char p.
Asymptotic results for average sizes of isogeny Selmer groups of hyperelliptic Jacobians are obtained by combining Bhargava geometry-of-numbers with new Vinberg-theory parametrizations for Dynkin types B and C, plus lower bounds via Greenberg-Wiles.
Constructs an equivalence for torsion coefficients between Zhu's category and Fargues-Scholze's category via Scholze's analytification functor and kimberlite theory, with applications to BunG decompositions and local Shimura varieties.
The condensed fundamental group of Spec(Z) is non-trivial, hence Spec(Z) is not condensed contractible.
For the standard representation of Sp_{2n}(C), the Gaiotto locus is the Bialynicki-Birula closure associated to U(Sp_{2n-2}(C)) inside the nilpotent cone, and its intersection with the stable cotangent chart is the closure of the conormal bundle to the one-spinor stratum of the generalized theta-div
The nth Amitsur group is a stable G-birational invariant of smooth projective G-varieties over char-0 fields for all n≥2.
Assuming Lang-Trotter-type sparsity for simultaneous supersingular reductions, the paper proves two Zilber-Pink-type finiteness results for Hodge generic curves in Y(1)^n via André's G-function method.
Lang-Trotter conjecture for elliptic curve pairs implies new Zilber-Pink cases for curves in A_3 via André's G-functions method, without boundary intersection assumptions.
A new fibration theorem implies solvable descent, solving the Grunwald problem for solvable groups up to the Brauer-Manin obstruction.
Under standard Taylor-Wiles hypotheses, every irreducible 2-dimensional totally odd mod p Galois representation of the absolute Galois group of a totally real field F admits lifts on arbitrary prescribed components of local deformation rings, allowing potentially semistable conditions with arbitrary
Shape theory for condensed anima recovers classical shape for paracompact compactly generated and locally contractible spaces while extending sheaf-condensed cohomology comparisons.
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A new perspective on the rank of Mazur's Eisenstein Hecke algebra
For primes N and p with N ≡ 1 mod p, the rank r of Mazur's Eisenstein Hecke algebra equals one plus the vanishing order of a mod-p zeta element interpolating L-values at -1 when r is 2 or 3, with a uniform extension to level N² and partial results for higher ranks.