Rank-2 attractors and Deligne’s conjecture

In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits. We will develop methods to analyze the algebraic de Rham cohomologies of rank-2 attractors, and we will illustrate how our methods work by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. We will look at the interesting connections between rank-2 attractors in string theory and Deligne’s conjecture on the special values of L-functions. We will also formulate several open questions concerning the potential connections between attractors in string theory and number theory.

WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT -FUNCTIONS

Let${\mathcal{K}}$be an imaginary quadratic field. Let${\rm\Pi}$and${\rm\Pi}^{\prime }$be irreducible generic cohomological automorphic representation of$\text{GL}(n)/{\mathcal{K}}$and$\text{GL}(n-1)/{\mathcal{K}}$, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called aWhittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if${\rm\Pi}$is cuspidal and the weights of${\rm\Pi}$and${\rm\Pi}^{\prime }$are in a standard relative position, the critical values of the Rankin–Selberg product$L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$are essentially algebraic multiples of the product of the Whittaker periods of${\rm\Pi}$and${\rm\Pi}^{\prime }$. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal${\rm\Pi}$can be given a motivic interpretation, and can also be related to a critical value of the adjoint$L$-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint$L$-functions are compatible with Deligne’s conjecture.