These notes give a statement of the fundamental lemma, which is a conjectural identity between p-adic integrals.

1. Iwasawa functions, p-adic measures, and distributions 1 1.1. Some preliminaries 1 1.2. Iwasawa functions and the Iwasawa algebra 3 1.3. Iwasawa functions and interpolation 4 1.4. p-adic distributions and p-adic measures 8 2. The p-adic L-functions of Dirichlet characters 10 2.1. Preliminaries 10 2.2. Iwasawa’s construction 12 3. The p-adic L-functions of modular forms 15 3.1. Preliminaries 1...

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 orPGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3. This corresponden...

In this paper, we have extended previously known human migration models and realized a series of numerical experiments on a model example to simulate the influence of the marginal migration costs, utility function values and the influence coefficients on the equilibrium states. The introduced influence coefficients allow one to consider and study not only the two classical models by Walras (per...

In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime p>3, show that∑k=0p−1(4k2k+1)(2kk)48k≡0(modp2), and∑k=0p−1(2kk)(3kk)24k≡{((2p−2)/3(p−1)/3)(modp2)ifp≡1(mod3),p/((2p+2)/3(p+1)/3)(modp2)ifp≡2(mod3). We also obtain some other results of such types.

We formulate a dynamic facility location model for a …rm locating on a discrete network. It is assumed that this locating …rm will act as the leader …rm in an industry characterized by Stackelberg leader-follower competition. The …rm’s I competitors are assumed to act as Cournot …rms and are each assumed to operate under the assumption of zero conjectural variation with respect to their I 1 Cou...

We state a conjectural relationship between the fermionic form [HKOTY] and the Betti numbers of a Grassmannian over a preprojective algebra or, equivalently , of a lagrangian quiver variety.