Jayce Getz

A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms

Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series E k in the standard fundamental domain for Γ lie on A:= {e iθ : π/2 ≤ θ ≤ 2π/3). In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique gap function in M k , the modular form with the maximal number of consecutive zero coefficients in its q-expansion following the constant 1, has zeros only on A. In addition, we show that the j-invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight k.

ON CONGRUENCE PROPERTIES OF THE PARTITION FUNCTION

Some congruence properties of the partition function are proved.

Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change

Chapter 1. Introduction.- Chapter 2. Review of Chains and Cochains.- Chapter 3. Review of Intersection Homology and Cohomology.- Chapter 4. Review of Arithmetic Quotients.- Chapter 5. Generalities on Hilbert Modular Forms and Varieties.- Chapter 6. Automorphic vector bundles and local systems.- Chapter 7. The automorphic description of intersection cohomology.- Chapter 8. Hilbert Modular Forms with Coefficients in a Hecke Module.- Chapter 9. Explicit construction of cycles.- Chapter 10. The full version of Theorem 1.3.- Chapter 11. Eisenstein Series with Coefficients in Intersection Homology.- Appendix A. Proof of Proposition 2.4.- Appendix B. Recollections on Orbifolds.- Appendix C. Basic adelic facts.- Appendix D. Fourier expansions of Hilbert modular forms.- Appendix E. Review of Prime Degree Base Change for GL2.- Bibliography.

Partition identities and a theorem of Zagier

In the study of partition theory and q-series, identities that relate series to infinite products are of great interest (such as the famous Rogers-Ramanujan identities). Using a recent result of Zagier, we obtain an infinite family of such identities that is indexed by the positive integers. For example, if m = 1, then we obtain the classical Eisenstein series identity Σλ≥1 odd (-1)(λ-1)/2qλ/(1-q2λ)=q Πn=1∞ (1-q8n)4/(1-q4n)2 If m = 2 and (·/3;) denotes the usual Legendre symbol modulo 3, then we obtain Σλ ≥1 (λ/3) qλ/(1-q2λ=q Πn=1∞ (1-qn)(1-q6n)6/(1-q2n)2(1-q3n)3 We describe some of the partition theoretic consequences of these identities. In particular, we find simple formulas that solve the well-known problem of counting the number of representations of an integer as a sum of an arbitrary number of triangular numbers.

A general simple relative trace formula

In this paper we prove a relative trace formula for all pairs of connected algebraic groups H G G, with G a reductive group and H the direct product of a reductive group and a unipotent group, given that the test function satisfies simplifying hypotheses. As an application, we prove a relative analogue of the Weyl law, giving an asymptotic formula for the number of eigenfunctions of the Laplacian on a locally symmetric space associated to G weighted by their L 2 -restriction norm over a locally symmetric subspace associated to H0 G.

Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields: SECONDARY TERMS FOR QUADRATIC FORMS

Let

The Full Version of Theorem 1.3

Q

Eisenstein Series with Coefficients in Intersection Homology

be a nondegenerate quadratic form on a vector space

Hilbert Modular Forms with Coefficients in a Hecke Module

V

Review of Chains and Cochains

of even dimension