The Darmon-Dasgupta units over genus fields and the Shimura correspondence

We study a special case of the Gross-Stark conjecture (Gross, 1981 [Gr]), namely over genus fields. Based on the same idea we provide evidence of the rationality conjecture of the elliptic units for real quadratic fields over genus fields, which is a refinement of the Gross-Stark conjecture given by Darmon and Dasgupta (2006) [DD]. Then a relationship between these units and the Fourier coefficients of p-adic Eisenstein series of half-integral weight is explained.     

L'opera scientifica di Gaetano Scorza

Discorso tenuto daF. Severi il 6 febbraio 1941-(XIX) presso la R. Università di Roma, per iniziativa della Facoltà di Scienze Fisiche Matematiche e Naturali, del Reale Istituto Nazionale di Alta Matematica e del Comitato per la Fisica e la Matematica Applicata del Consiglio Nazionale delle Ricerche.     

Scorza varieties and Jordan algebras

In this paper, I give two very direct proves of the correspondance between a geometric object (Scorza varieties) and an algebraic one (Jordan algebras). I also give a short proof of the homogeneity of Scorza varieties, and a new and very simple proof of properties of the automorphism group of a Jordan algebra.     

The Local gabriel correspondence

The first part of the present work introduces a new notion of ring of quotients by any filter of one sided ideals. The construction is a generalization of the one by Gabriel which works for idempotent filters. The second part is devoted to an extension of crossed action of Hopf algebras to such a ring of quotients. The last part comprises an application of this machinery to the question about crossed products being semiprime Goldie.     

The problem of boundary correspondence

It is proved that those points of the boundary of the disk, to which a homeomorphism of the class BLP, cannot be extended according to Carathéodory (from the disk), form a set of zero (p-)-capacity. Estimates for the distortion of distances under these mappings are given.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 103–108, July, 1977.     

Singular Poisson-Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.     

The theta correspondence for similitudes

In this paper we investigate the theta correspondence for similitudes over a nonarchimedean local field. We show that the two main approaches to a theta correspondence for similitudes from the literature are essentially the same, and we prove that a version of strong Howe duality holds for both constructions. During the period of this work the author was a Research Associate with the NSF 1992–1994 special projectTheta Functions, Dual Pairs, and Automorphic Forms at the University of Maryland, College Park     

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.     

The maximum genus of a 3-regular simplicial graph

In this paper, the relationship between non-separating independent number and the maximum genus of a 3-regular simplicial graph is presented. A lower bound on the maximumgenus of a 3-regular graph invalving girth is provided. The lower bound is tight, it improves a bound of Huang and Liu.     

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g =  3. Furthermore, we determine the part of the orbifold cohomology of the Deligne–Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of ${\mathcal{M}3}$ .     

The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S ₃ onto a genus 2 Riemann surface S ₂ by Hol(S ₃, S ₂). Call two mappings f and g in Hol(S ₃, S ₂) equivalent whenever there exist conformal automorphisms α and β of S ₃ and S ₂ respectively with f ? α = β ? g. It is known that Hol(S ₃, S ₂) always consists of at most two equivalence classes.We obtain the following results: If Hol(S ₃, S ₂) consists of two equivalence classes then both S ₃ and S ₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S ₃, S ₂) there exist anticonformal automorphisms α? and β? with f ? α? = β? ? g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S ₃, S ₂) such that Hol(S ₃, S ₂) consists of two equivalence classes.     

The coherent-constructible correspondence and Fourier-Mukai transforms

As evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In a previous paper, the authors showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on ℝ ⁿ . In this paper we investigate and reprove some of Kawamata’s results from this perspective.