The modular properties of generalized theta-functions with characteristics are used to build cusp form corresponding to quadratic forms in ten variables.
As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinit...As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.展开更多
We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed...We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.展开更多
We study the Gross conjecture for the cyclotomic function field extension k(∧f)/k where k = Fq(t) is the rational function field and f is a monic polynomial in Fq[t].We prove the conjecture in the Fermat curve case(i...We study the Gross conjecture for the cyclotomic function field extension k(∧f)/k where k = Fq(t) is the rational function field and f is a monic polynomial in Fq[t].We prove the conjecture in the Fermat curve case(i.e., when f = t(t - 1)) by a direct calculation. We also prove the case when f is irreducible, which is analogous to the Weil reciprocity law. In the general case, we manage to show the weak version of the Gross conjecture here.展开更多
文摘The modular properties of generalized theta-functions with characteristics are used to build cusp form corresponding to quadratic forms in ten variables.
基金The author was supported by Labex CEMPI(Centre Europeen pour les Mathematiques,la Physique et leurs Interaction).
文摘As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.
文摘We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.
基金supported by the National Natural Sciencc Foundation of China(Grant No.10401018)the Scientifc Research Foundation for the Returmed Overseas Chinese Scholars,State Education Ministy.
文摘We study the Gross conjecture for the cyclotomic function field extension k(∧f)/k where k = Fq(t) is the rational function field and f is a monic polynomial in Fq[t].We prove the conjecture in the Fermat curve case(i.e., when f = t(t - 1)) by a direct calculation. We also prove the case when f is irreducible, which is analogous to the Weil reciprocity law. In the general case, we manage to show the weak version of the Gross conjecture here.
