arXiv subjects

Tuoping Du

Publications and source records attributed to Tuoping Du.

9 recordsLinked to original sources

arXiv2026

The triality of the twisted discrete trace formula for PGSO(8)

In this paper, we establish the triality twisted trace formula for PGSO(8), including its discrete part, and obtain a coarse classification of its automorphic representations by combining the properties of triality. By comparing the standard trace formula for G_2 with the triality twisted trace formula for PGSO(8), we derive a corresponding coarse classification for automorphic representations of G_2. Specifically, we construct the triality-twisted elliptic endoscopic data for PGSO(8), and the elliptic endoscopic data for G_2. Based on these constructions and the general framework of trace formulas, we establish the relevant trace formulas. Utilizing the triality property of PGSO(8), we obtain a coarse classification of its automorphic representations, which in turn yields a coarse classification for those of G_2.

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arXiv2026

On the central derivatives of l-functions and modularity of heenger cycles

This paper establishes an arithmetic intersection formula for central L-derivatives in higher weights.We prove that for a general cusp form (extending the previous result for newforms), the derivative is represented by the global height pairing between higher Heegner cycles. This result provides a framework for the Gross-Zagier-Zhang formula and its generalizations.Furthermore, we investigate the modularity of the generating series of Heegner cycles,proving a weak version of the conjecture and reducing the full modularity to a vanishing conjecture,for which we provide supporting evidence.

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arXiv2024

On the arithmetic inner product formula and central derivatives of L-functions

This research provides a formal definition of the arithmetic theta lift for cusp forms of weight $3/2$ and establishes the arithmetic inner product formula, thereby completing the Kudla program on modular curves. This formula is demonstrated to be equivalent to the Gross-Zagier formula, for which we provide a new proof. Additionally, the authors introduce a new arithmetic representation for the central derivatives of L-functions associated with cusp forms of higher weight. Although this representation differs from Zhang's higher weight Gross-Zagier formula, it maintains a significant connection to it. This study also proposes a conjecture indicating that the vanishing of derivatives of L-functions is determined by the algebraicity of the coefficients of harmonic weak Maass forms. A consistent approach is employed to study both parts of this work.

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arXiv2024

An exact way to verify whether n is a congruent number using Heegner points

We introduce the relationship between congruent numbers and elliptic curves, and compute the conductor of the elliptic curve $y^2 = x^3 - n^2 x$ associated with it. Furthermore, we prove that its $L$-series coefficient $a_m = 0$ when $m \equiv 3 \mod 4$.By using the invariants of the elliptic curve introduced above, we calculate Heegner points to quickly verify whether $n$ is a congruent number.

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arXiv2018

Weak Siegel-Weil formula for M_2(Q) and arithmetic on quaternions

In this paper, we prove the weak Siegel-Weil formula for the space M_2(Q) . We study the Hecke correspondence and representation numbers associated to Eichler orders, and give the explicit formula for degree of Hecke correspondence and average representation numbers over genus. This formula could recover the main results in [DuYang]. We could identify these numbers with the Fourier coefficients of Eisenstein series via Siegel-Weil formula(weak Siegel-Weil formula for M_2(Q). In the last part of this article, we reprove four square sum Theorem via Siegel-Weil formula and Kudla's matching.

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arXiv2018

Twisted arithmetic Siegel Weil formula on X0(N)

In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (\Delta, r) where \Delta >0 and \Delta \equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function \phi_{\Delta, r}(\tau) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of \phi_{\Delta, r}(\tau), along the way, we also identify the arithmetic pairing \langle \phi_{\Delta, r}(\tau),\widehat{\omega}_N \rangle with special value of some Eisenstein series, where \widehat{\omega}_N is a normalized metric Hodge line bundle.

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arXiv2017

Arithmetic Siegel-Weil formula on $X_{0}(N)$

In this paper, we proved an arithmetic Siegel-Weil formula and the modularity of some arithmetic theta function on the modular curve $X_0(N)$ when $N$ is square free. In the process, we also construct some generalized Delta function for $\Gamma_0(N)$ and proved some explicit Kronecker limit formula for Eisenstein series on $X_0(N)$.

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arXiv2014

Ternary quadratic forms and Heegner divisors

In this paper, I use Siegel-Weil formula and Kudla matching principle to prove some interesting identities between representation number (of ternary quadratic space) and the degree of Heegner divisors.

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arXiv2012

Quaternions and Kudla's matching princple

In this paper, we prove some interesting identities, among average representation numbers (associated to definite quaternion algebras) and `degree' of Hecke correspondences on Shimura curves (associated to indefinite quaternion algebras).

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