D4 Modular Forms

Estimates of triple products of automorphic functions II

We prove a sharp bound for the average value of the triple product of modular functions for the Hecke subgroup Γ_0(N). Our result is an extension of the main result in {Bernstein&Reznikov-2004} to a fixed cuspidal representation of the adele group PGL(2,A).

计算 $G_2$ 上的离散、Level-$1$ 四元数自守表示

Counting Discrete, Level-$1$, Quaternionic Automorphic Representations on $G_2$

本文利用“hyperendoscopy”技术发展一般迹公式，研究例外群 $G_2$ 上的四元数自守表示，克服了离散级数非正则的技术困难，并计算出 level-$1$ 表示空间的维数。

Poincaré series and modular functions for U(n, 1)

We construct infinitely many nonholomorphic automorphic forms and modular forms associated to a discrete subgroup of infinite covolume of $U(n, 1)$.

A Generalization of Level-Raising Congruences for Algebraic Modular Forms

In this paper we prove a general theorem about congruences between automorphic forms on a reductive group G which is compact at infinity modulo the center. If the rank is one, this essentially reduces to Ribet's level-raising theorem. We then specialize to the higher rank case where G is an inner form of GSp(4). Here we get congruences with automorphic forms having a generic local component. In particular, a Saito-Kurokawa form is congruent to a form which is not of Saito-Kurokawa type. We get similar results for U(3) at split primes.

Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula

We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we e…

Rank 2 symmetric hyperbolic Kac-Moody algebras and Hilbert modular forms

In this paper we study rank two symmetric hyperbolic Kac-Moody algebras H(a) and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(p) for each odd prime p and show that there exists a chain of embeddings in each family. When p = 5, 13, 17, we show that the first H(a) in each family, i.e. H(3), H(11), H(66), is contained in a generalized Kac-Moody superalgebra whose denominator function is a Hilbert modular form given by a Borcherds product. Hence, our results provide automorphic correction for those H(a)'s. We also compute …

The completed standard $L$-function of modular forms on $G_2$

The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(π,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $π$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $π$ of $G_2(\mathbb{A})$ that corresponds to a modular form $φ$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $Λ(π,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $φ$ is nonzero, we prove the functional equation $Λ(π,\operatorname{Std},s) = Λ(π,\operatorname{Std},1-s)$.…

The Largest Mathieu Group and (Mock) Automorphic Forms

We review the relationship between the largest Mathieu group and various modular objects, including recent progress on the relation to mock modular forms. We also review the connections between these mathematical structures and string theory on K3 surfaces.

Level lowering for GU(1,2)

Mazur's principle gives a criterion under which an irreducible mod $\ell$ Galois representation arising from a modular form of level $Np$ (with $p$ prime to $N$) can also arise from a modular form of level $N.$ We prove an analogous result showing that a mod $\ell$ Galois representation arising from a stable cuspidal automorphic representation of the unitary similitude group $G=\mathrm{GU}(1,2)$ which is Steinberg at an inert prime $p$ can also arise from an automorphic representation of $G$ that is unramified at $p$.

Siegel modular forms of degree two attached to Hilbert modular forms

Let E/Q be a real quadratic field and pi_0 a cuspidal, irreducible, automorphic representation of GL(2,A_E) with trivial central character and infinity type (2,2n+2) for some non-negative integer n. We show that there exists a non-zero Siegel paramodular newform F with weight, level, Hecke eigenvalues, epsilon factor and L-function determined explicitly by pi_0. We tabulate these invariants in terms of those of pi_0 for every rational prime p.

The Spin $L$-function on $\mathrm{GSp}_6$ for Siegel modular forms

We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$-function on $\mathrm{PGSp}_6$. The integral applies to the cuspidal automorphic representations associated to Siegel modular forms. If $π$ corresponds to a level one Siegel modular form $f$ of even weight, and if $f$ has a non-vanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $Λ(π,Spin,s)$ of $π$.

关于在实位生成大离散序列表示的$\operatorname{Mp}_4(\mathbb{A}_\mathbb{Q})$与$\operatorname{SO}_5(\mathbb{A}_\mathbb{Q})$上自守形式的Ibukiyama对应

Ibukiyama correspondences on automorphic forms on $\operatorname{Mp}_4(\mathbb{A}_\mathbb{Q})$ and $\operatorname{SO}_5(\mathbb{A}_\mathbb{Q})$ generating large discrete series representations at the real place

本文针对实位生成大离散序列表示的Mp₄(A_Q)与SO₅(A_Q)上的自守形式，建立并证明了Ibukiyama对应，同时阐明该对应可通过局部theta对应来描述。

Explicit Constructions of Automorphic Forms: Theta Correspondence and Automorphic Descent

This is an expanded set of notes based on two lectures given by the author at the 2022 IHES summer school on the Langlands program, on explicit techniques for the construction of automorphic representations.

Irreducibility Criteria for Local and Global Representations

It is proved that certain types of modular cusp forms generate irreducible automorphic representation of the underlying algebraic group. Analogous archimedean and non-archimedean local statements are also given.

24 faces of the Borcherds modular form Phi_{12}

The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebra…

关于Picard模曲面的高阶调节子

On higher regulators of Picard modular surfaces

本文证明Picard模曲面的动机类位于内部动机上同调群中，并建立其与动机L函数非临界值的关系，从而验证Beilinson猜想。

Rademacher sums, moonshine and gravity

In 1939 Rademacher derived a conditionally convergent series expression for the elliptic modular invariant, and used this expression- the first Rademacher sum - to verify its modular invariance. By generalizing Rademacher's approach we construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for all groups commensurable with the modular group. We use these Rademacher sums to illuminate various aspects of the structure of the spaces of automorphic integrals, including the actions of Hecke operators. We obtain a new characterization of the discrete groups of mons…