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We introduce a Frobenius-like structure for the [Formula: see text] Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind. 

We consider the KZ equations over C in the case, when the hypergeometric solutions are hyperelliptic integrals of genus g. Then the space of solutions is a 2g-dimensional complex vector space. We also consider the same equations modulo ps, where p is an odd prime and s is a positive integer, and over the field Q_p of p-adic numbers. We construct polynomial solutions of the KZ equations modulo ps and study the space Mps of all constructed solutions. We show that the p-adic limit of Mps as s→∞ gives us a g-dimensional vector space of solutions of the KZ equations over Qp. The solutions over Qp are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the case g=1 of elliptic integrals. The p-adic limit of Mps as s→∞ gives us a one-dimensional space of solutions over Qp at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over Qp defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Qp. We describe the Frobenius transformations of solutions of the KZ equations for g=1 and then recover the unit roots of the zeta functions of the elliptic curves defined by the equations y2=βx(x−1)(x−α) over the finite field Fp. Here α,β∈F×p,α≠1 

We consider the master functions associated with one irreducible integrable highest weight representation of a Kac-Moody algebra. We study the generation procedure of new critical points from a given critical point of one of these master functions. We show that all critical points of all these master functions can be generated from the critical point of the master function with no variables. In particular this means that the set of all critical points of all these master functions form a single population of critical points. We formulate a conjecture that the number of populations of critical points of master functions associated with a tensor product of irreducible integrable highest weight representations of a Kac-Moody algebra are labeled by homomorphisms to $$\C$$ of the Bethe algebra of the Gaudin model associated with this tensor product. 

We consider the KZ differential equations over C in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field F_p. We study the polynomial solutions of these differential equations over F_p, constructed in a previous work joint with V. Schechtman and called the F_p-hypergeometric solutions. The dimension of the space of F_p-hypergeometric solutions depends on the prime number p. We say that the KZ equations have ample reduction for a prime p, if the dimension of the space of F_p-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over C. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis F_p-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials (z_i−z_j)^{M_i+M_j} are replaced with (z_i−z_j)^{Mi+Mj−p} and the Euler gamma function Γ(x) is replaced with a suitable F_p-analog defined on F_p