Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.

We define the n-point function for a vertex operator algebra on a genus two Riemann surface in two separate sewing schemes where either two tori are sewn together or a handle is sewn to one torus. We explicitly obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and lattice vertex operator algebras in both sewing schemes. We prove that the partition functions are holomorphic in the sewing parameters on given suitable domains and describe their modular properties. Finally, we show that the partition functions cannot be equal in the neighborhood of a two-tori degeneration point where they can be explicitly compared.

A Course in Arithmetic

The Genus Two Partition Function for Free Bosonic and Lattice Vertex Operator Algebras

https://www.ams.org/tran/1990-321-01/S0002-9947-1990-0986684-6/S0002-9947-1990-0986684-6.pdf

The 27-dimensional module for ₆. III

Splitting of hermitian forms over group rings

We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.

background

Accessory parameters for Liouville theory on the torus

Abstract We use arithmetic models of modular curves to establish some properties of Ramanujan's continued fraction. In particular, we give a new geometric proof that its singular values are algebraic units that generate specific abelian extensions of imaginary quadratic fields, and we use a mixture of geometric and analytic methods to construct and study an infinite family of two-variable polynomials over ℤ that are related to Ramanujan's function in the same way that the classical modular polynomials are related to the classical j-function. We also prove that a singular value on the imaginary axis, necessarily real, lies in a radical tower in ℝ only if all odd prime factors of its degree over ℚ are Fermat primes; by computing some ray class groups, we give many examples where this necessary condition is not satisfied.

A Course in Arithmetic

Citations

The Genus Two Partition Function for Free Bosonic and Lattice Vertex Operator Algebras

The 27-dimensional module for ₆. III

Splitting of hermitian forms over group rings

Accessory parameters for Liouville theory on the torus

Modular curves and Ramanujan's continued fraction

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