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Let and let and be two line bundles on . Consider the cup-product map
where and . We answer two natural questions about the map above: When is it a nonzero homomorphism of representations of ? Conversely, given generic irreducible representations and , which irreducible components of may appear in the right hand side of the equation above? For the first question we find a combinatorial condition expressed in terms of inversion sets of Weyl group elements. The answer to the second question is especially elegant: the representations appearing in the right hand side of the equation above are exactly the generalized PRV components of of stable multiplicity one. Furthermore, the highest weights corresponding to the representations fill up the generic faces of the Littlewood–Richardson cone of of codimension equal to the rank of . In particular, we conclude that the corresponding Littlewood–Richardson coefficients equal one.
A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper, the authors observed that in a particular example two total masses coming from two different weightings are dual to each other. We discuss the problem of how generally such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincaré duality for stringy invariants. We also exhibit several examples.
Let be a complete, algebraically closed nonarchimedean valued field, and let have degree . We show there is a canonical way to assign nonnegative integer weights to points of the Berkovich projective line over in such a way that . When has bad reduction, the set of points with nonzero weight forms a distributed analogue of the unique point which occurs when has potential good reduction. Using this, we characterize the minimal resultant locus of in analytic and moduli-theoretic terms: analytically, it is the barycenter of the weight-measure associated to ; moduli-theoretically, it is the closure of the set of points where has semistable reduction, in the sense of geometric invariant theory.
The point of this paper is to give an explicit -adic analytic construction of two Iwasawa functions, and , for a weight-two modular form and a good prime . This generalizes work of Pollack who worked in the supersingular case and also assumed . These Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: we bound the rank and estimate the growth of the Šafarevič–Tate group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.
Artin solved Hilbert’s 17th problem, proving that a real polynomial in variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only squares are needed.
In this paper, we investigate situations where Pfister’s theorem may be improved. We show that a real polynomial of degree in variables that is positive semidefinite is a sum of squares of rational functions if . If is even or equal to or , this result also holds for .
Let be the Möbius function and let . We prove that the Gowers -norm of restricted to progressions is on average over for any , where is an arbitrary residue class with . This generalizes the Bombieri–Vinogradov inequality for , which corresponds to the special case .
We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. Thanks to this analysis, we obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus. We spell out this bound in several examples, and we use it to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.