A symplectic restriction problem

We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods.

ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree $3$ and level $1$ .

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: The structure of the invariant

Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.

The Modularity of Special Cycles on Orthogonal Shimura Varieties over Totally Real Fields under the Beilinson–Bloch Conjecture

We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree d associated with a quadratic form in $$n+2$$ variables whose signature is $$(n,2)$$ at e real places and $$(n+2,0)$$ at the remaining $$d-e$$ real places for $$1\leq e <d$$ . Recently, these cycles were constructed by Kudla and Rosu–Yott, and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson–Bloch conjecture on the injectivity of the higher Abel–Jacobi map, the generating series of special cycles of codimension er in the Chow group is a Hilbert–Siegel modular form of genus r and weight $$1+n/2$$ . Our result is a generalization of Kudla’s modularity conjecture, solved by Yuan–Zhang–Zhang unconditionally when $$e=1$$ .

Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

Motivated by precision counting of BPS black holes, we analyze six-derivative couplings in the low energy effective action of three-dimensional string vacua with 16 supercharges. Based on perturbative computations up to two-loop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the \nabla^2(\nabla\phi)^4∇2(∇ϕ)4 effective interaction is given by a genus-two modular integral of a Siegel theta series for the non-perturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including tree-level, one- and two-loop corrections, plus non-perturbative effects of order e^{-1/g_3^2}e−1/g32. We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results, . In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact \nabla^2 F^4∇2F4 effective interaction in four-dimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order e^{-R}e−R from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order e^{-R^2}e−R2 from Taub-NUT instantons. We show that instanton corrections from 1/4-BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract two-instanton corrections from pairs of 1/2-BPS black holes, demonstrating consistency with supersymmetry and wall-crossing, and estimate the size of instanton-anti-instanton contributions.

Weights of the Mod p Kernels of Theta Operators

Let Θ[j] be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime p, we give the weights of elements of mod p kernel of Θ[j], where the mod p kernel of Θ[j] is the set of all Siegel modular forms F such that Θ[j](F) is congruent to zero modulo p. In order to construct examples of the mod p kernel of Θ[j] fromany Siegel modular form, we introduce new operators A(j)(M) and show the modularity of F|A(j)(M) when F is a Siegel modular form. Finally, we give some examples of the mod p kernel of Θ[j] and the filtrations of some of them.

The spin -function on for Siegel modular forms

We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$-function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of $\unicode[STIX]{x1D70B}$.

On mod p singular modular forms

We show that a Siegel modular form with integral Fourier coefficients in a number field

The non-existence of stable Schottky forms

We show that there is no stable Siegel modular form that vanishes on every moduli space of curves.