Zagier Theorem Research Articles

The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem

Abstract We relate p-adic families of Jacobi forms to big Heegner points constructed by B. Howard, in the spirit of the Gross–Kohnen–Zagier theorem. We view this as a GL ⁢ ( 2 ) {\mathrm{GL}(2)} instance of a p-adic Kudla program.

Alternating double Euler sums, hypergeometric identities and a theorem of Zagier

In this work, we derive relations between generating functions of double stuffle relations and double shuffle relations to express the alternating double Euler sums ζ(r‾,s), ζ(r,s‾) and ζ(r‾,s‾) with r+s odd in terms of zeta values. We also give a direct proof of a hypergeometric identity which is a limiting case of a basic hypergeometric identity of Andrews. Finally, we gave another proof for the formula of Zagier on the multiple zeta values ζ(2,…,2,3,2,…,2).

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

An alternative proof of a theorem of Zagier

We give a new proof of Zagier's theorem on evaluation of multiple zeta (star) values at ({2}a,3,{2}b). The novelty of the proof consists in employing a hypergeometric identity of Andrews, which expresses a very-well-poised hypergeometric series in terms of a multiple series, and the residue theorem.

On the $p$-converse of the Kolyvagin–Gross–Zagier theorem

Let A / \mathbb Q be an elliptic curve having split multiplicative reduction at an odd prime p . Under some mild technical assumptions, we prove the statement: \mathrm {rank}_{\mathbb Z} A(\mathbb Q)=1\ \text{and}\ \# (\mathrm {Ш}(A / \mathbb Q)_{p^{\infty}}\big)<\infty\ \ \implies\ \ \mathrm{ord}_{s=1}L(A / \mathbb Q,s)=1, thus providing a ‘ p -converse’ to a celebrated theorem of Kolyvagin–Gross–Zagier.

A Gross–Zagier formula for quaternion algebras over totally real fields

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factoriza- tion of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields K and K ' , when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in Bp,∞, the definite quaternion algebra over Q ramified only at p and infinity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM fields into superspecial orders in definite quaternion algebras over totally real fields of strict class number 1. Our results can also be viewed as a counting formula for the number of isomorphisms modulo p|p between abelian varieties with CM by different fields. Our counting formula can also be used to determine which superspecial primes appear in the factorizations of differences of values of Siegel modular functions at CM points associated to two different CM fields, and to give a bound on those supersingular primes which can appear. In the special case of Jacobians of genus 2 curves, this provides information about the factorizations of numerators of Igusa invariants, and so is also relevant to the problem of constructing genus 2 curves for use in cryptography.

Arithmetic of higher dimensional Dedekind–Rademacher sums

Text In this paper we investigate higher order dimensional Dedekind–Rademacher sums given by the expression 1 a 0 m 0 + 1 ∑ k = 1 a 0 − 1 ∏ j = 1 d cot ( m j ) ( π a j k a 0 ) , where a 0 is a positive integer, a 1 , … , a d are positive integers prime to a 0 and m 0 , m 1 , … , m d are non-negative integers. We study arithmetical properties of these sums. For instance, we prove that these sums are rational numbers and we explicit good bounds for their denominators. A reciprocity law is given generalizing a theorem of Rademacher for the classical Dedekind sums and a theorem of Zagier for higher dimensional Dedekind–Rademacher sums. On the other hand, our reciprocity results can be viewed as complements to the Beck reciprocity theorem. Taking m 0 = ⋯ = m d = 0 , we derive the reciprocity and rationality theorems of Zagier. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=J1_5H28fgAg.

Higher-Weight Heegner Points

In this paper we formulate a conjecture that partially generalizes the Gross–Kohnen–Zagier theorem to higher-weight modular forms. For ƒ ∊ S 2k (N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus /L ƒ defined by ƒ. We define higher-weight analogues of Heegner divisors on /L ƒ. We conjecture that they all lie on a line and that their positions are given by the coefficients of a certain Jacobi form corresponding to ƒ. In weight 2, our map is the modular parameterization map (restricted to Heegner points), and our conjectures are implied by Gross–Kohnen–Zagier. For any weight, we expect that our map is the Abel–Jacobi map on a certain modular variety, and so our conjectures are consistent with the conjectures of Beilinson–Bloch. We have verified that our map is the Abel–Jacobi map for weight 4. We provide numerical evidence to support our conjecture for a variety of examples.

The Gross–Kohnen–Zagier theorem over totally real fields

AbstractOn Shimura varieties of orthogonal type over totally real fields, we prove a product formula and the modularity of Kudla’s generating series of special cycles in Chow groups.

The Iwasawa theoretic Gross–Zagier theorem

We prove Mazur and Rubin's Iwasawa-theoretic Gross-Zagier conjecture (under some restrictive hypotheses), which relates Heegner points in towers of number fields to the 2-variable p-adic L-function. The result generalizes Perrin-Riou's p-adic Gross-Zagier theorem.

Evaluation of double L-values

In this paper, a theorem of Zagier concerning double zeta evaluations is generalized to the double L-values. In addition, fast computation of the double L-values is demonstrated, extending the method of Crandall. The PARI commands are available electronically.

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 ∑ λ⩾1 odd (−1) (λ−1)/2q λ (1−q 2λ) =q ∏ n=1 ∞ (1−q 8n) 4 (1−q 4n) 2 . If m=2 and ( · 3 ) denotes the usual Legendre symbol modulo 3, then we obtain ∑ λ⩾1 ( λ 3 )q λ (1−q 2λ) =q ∏ n=1 ∞ (1−q n)(1−q 6n) 6 (1−q 2n) 2(1−q 3n) 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.

TheGL(2) Rankin–Selberg Method for Functions Not of Rapid Decay

In this note, we give a new proof of Zagier's theorem concerning the Rankin–Selberg method for functions not of rapid decay.