Frobenius Structure Research Articles

Descente par Frobenius Explicite pour les [formula omitted]†-Modules

In his theory of D†X-modules, P. Berthelot has proved a general theorem of Frobenius descent. On the other hand, Christol and others have also proved various statements about weak Frobenius structures on an annulus of the rigid projective line. It is the aim of this paper to give some explicit and global formulas for Frobenius descent for D†X-modules. It requires us to build a new kind of differential operator which represents Dwork's ψ operator. The construction uses mainly Taylor series properties. It is possible to derive from this some new proofs of Christol's theorems; it is also useful for computations in characteristicplike those arising in the construction of Cartier's isomorphism or Cartier's operator.

Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions .

Polynomial solutions to the WDVV equations in four dimensions

All polynomial solutions of the WDVV equations for the case n = 4 are determined. We find all five solutions predicted by Dubrovin, namely those corresponding to Frobenius structures on orbit spaces of finite Coxeter groups. Moreover we find two additional series of polynomial solutions of which one series is of semi-simple type (massive). This result supports Dubrovin's conjecture if modified appropriately.

Special values of symmetric hypergeometric functions

We discuss the p p -adic formula (0.3) of P. Th. Young, in the framework of Dwork’s theory of the hypergeometric equation. We show that it gives the value at 0 of the Frobenius automorphism of the unit root subcrystal of the hypergeometric crystal. The unit disk at 0 is in fact singular for the differential equation under consideration, so that it’s not a priori clear that the Frobenius structure should extend to that disk. But the singularity is logarithmic, and it extends to a divisor with normal crossings relative to Z p \mathbf {Z}_{p} in P Z p 1 \mathbf {P}^{1}_{\mathbf {Z}_{p}} . We show that whenever the unit root subcrystal of the hypergeometric system has generically rank 1, it actually extends as a logarithmic F F -subcrystal to the unit disk at 0. So, in these optics, “singular classes are not supersingular”. If, in particular, the holomorphic solution at 0 is bounded, the extended logarithmic F F -crystal has no singualrity in the residue class of 0, so that it is an F F -crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form.

On when a separable field extension in characteristic p > 0 is determined by its Frobenius structure, II

On when a separable field extension in characteristic p > 0 is determined by its Frobenius structure, II

On when a separable field extension in characteristic p > 0 is determined by its Frobenius structure, I

Let G c Aut,( V) be a subgroup of automorphisms of a finite-dimensional vector space V over a field k. A vector u E I’ is said to be a G-orbit basis generator for V if the set G(u) is a linear basis for I’. In this paper we continue the study begun in [Rl, R2] of the following question: If U, U are G-orbit basis generators for V then when are the stabilizers G, and G, conjugate? The conjugacy question is a more general formulation of a question concerning finite-dimensional separable field extensions E, F over an infinite field k of characteristic p > 0. Let k be a field of characteristic p > 0 and suppose E and F are n-dimensional separable field extensions of k. Recall that the associative structure of any commutative algebra A over k affords A a restricted Lie (Frobenius) structure Y(A) by [ab] =0 and aCpl =up for a, h E A. A natural question to ask about the pair E and F is the following: If Z(E) and Z(F) are isomorphic Lie algebras, then when are E and F isomorpic field extensions? Suppose P(E) N P(F). It does not necessarily follow that E = F;

A generalized Frobenius structure for coalgebras with applications to character theory

A generalized Frobenius structure for coalgebras with applications to character theory

On $p$-adic differential equations. The Frobenius structure of differential equations

On $p$-adic differential equations. The Frobenius structure of differential equations

The hypergeometric functions of the Faber–Zagier and Pixton relations

The relations in the tautological ring of the moduli space Mg of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space Mg,n of stable curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to P 1 ). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19 th century). Several open questions are proposed.