arXiv subjects

Zavosh Amir-Khosravi

Publications and source records attributed to Zavosh Amir-Khosravi.

5 recordsLinked to original sources

arXiv2020

Period Relations for Quaternionic Elliptic Functions

We study elliptic functions in quaternionic analysis, and prove some analogues of classical theorems from the complex case. The main result is a relation between the periods of closed differential 1-forms and 3-forms on H/L where L is a lattice in H. By applying to forms that are regular in the sense of quaternionic analysis, we obtain quaternionic analogues of Riemann's period relations for an abelian surface. We also obtain a quaternionic analogue of Legendre's relation, as an equality between $2π^2$ and a linear combination of the four quasi-periods of the quaternionic Weierstrass $ζ$-function, where the coefficients are rational functions of a basis for L. When L is a left-ideal for a maximal order in a definite quaternion algebra over Q, we obtain three further relations among the quasi-periods.

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arXiv2019

Fueter-Regular Discete Series for Sp(1,1)

We show that the quaternionic discrete series on G=Sp(1,1) with minimal K-type of dimension n+1 can be realized inside the space of Fueter-regular functions on the quaternionic ball B in H, with values in H^n. We then consider the corresponding $H^n$-valued Fueter-regular automorphic forms on G. For a fixed level $Γ$, we construct a non-trivial map from the space of pairs of such automorphic forms, to closed $M_n(H)$-valued differential $3$-forms on $B$, which transform under $Γ$ according to a cocycle condition.

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arXiv2019

Theta Functions and Reciprocity Laws

In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using a Möbius transform, and show they have a functional equation, Euler product factorization, and roots only on the critical line. In the second part, we prove a general reciprocity law for sums of exponentials of rational quadratic forms in any number of variables, using the transformation formula of the Riemann theta function on the Siegel upper half-space.

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arXiv2017

Generating series of intersection volumes of special cycles on unitary Shimura varieties

We form a generating series of regularized volumes of intersections of special cycles on a non-compact unitary Shimura variety with a fixed base change cycle. We show that it is a Hilbert modular form by identifying it with a theta integral, which we show converges even though the parameters lie outside the classical convergence range of Weil. By applying the regularized Siegel-Weil formulas of Ichino and Gan-Qiu-Takeda, we show the modular form is the restriction of a hermitian modular form of degree n related to Siegel Eisenstein series on U(n,n). An essential fact used is a computation showing the Kudla-Millson Schwartz function vanishes under the Ikeda map.

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arXiv2017

Serre's Tensor Construction and Moduli of Abelian Schemes

Given a polarized abelian scheme with action by a ring, and a projective finitely presented module over that ring, Serre's tensor construction produces a new abelian scheme. We show that to equip these abelian schemes with polarizations it's enough to equip the projective modules with non-degenerate positive-definite hermitian forms. As an application, we relate certain moduli spaces of principally polarized abelian schemes with action by the ring of integers of a CM field. More specifically, we consider integral models of zero-dimensional Shimura varieties associated to compact unitary groups. We show that all abelian schemes in such moduli spaces are, étale locally over their base schemes, Serre constructions of CM abelian schemes with positive-definite hermitian modules. We also describe the morphisms between such objects in terms of morphisms between the constituent data, and formulate these relations as an isomorphism of algebraic stacks.

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