Lambert series associated to Hilbert modular form

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

Non-tautological Hurwitz cycles

We show that various loci of stable curves of sufficiently large genus admitting degree d covers of positive genus curves define non-tautological algebraic cycles on $${\overline{{\mathcal {M}}}}_{g,N}$$ M ¯ g , N , assuming the non-vanishing of the d-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.

Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

Kim et al. (Proc Am Math Soc 144:687–3700, 2016) introduced the notion of odd-balance unimodal sequences in 2016. Like was shown by Bryson et al. (Proc Natl Acad Sci USA 109:16063–16067, 2012) for the generating function of strongly unimodal sequences, the generating function for odd-balanced unimodal sequences also has quantum modular behavior. Odd-balanced unimodal sequences thus appear to be a fundamental piece in the world of modular forms and combinatorics, and understanding their asymptotic properties is important for understanding their place in this puzzle. In light of this, we compute an asymptotic estimate for odd balanced unimodal sequences for ranks congruent to $$a \pmod {c}$$ a ( mod c ) for $$c\ne 2$$ c ≠ 2 or a multiple of 4. We find the interesting result that the odd balanced unimodal sequences are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which, are asymptotically related to the partition function. Our proofs of the main theorems rely on the representation of the generating function in question as a mixed mock modular form.

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 NONCRITICAL GALOIS REPRESENTATIONS

We propose a conjecture that the Galois representation attached to every Hilbert modular form is noncritical and prove it under certain conditions. Under the same condition we prove Chida, Mok and Park’s conjecture that Fontaine-Mazur L-invariant and Teitelbaum-type L-invariant coincide with each other.

Irreducibility of limits of Galois representations of Saito–Kurokawa type

We prove (under certain assumptions) the irreducibility of the limit $$\sigma _2$$ σ 2 of a sequence of irreducible essentially self-dual Galois representations $$\sigma _k: G_{{\mathbf {Q}}} \rightarrow {{\,\mathrm{GL}\,}}_4(\overline{{\mathbf {Q}}}_p)$$ σ k : G Q → GL 4 ( Q ¯ p ) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to $$1 \oplus \rho \oplus \chi $$ 1 ⊕ ρ ⊕ χ with $$\rho $$ ρ irreducible, two-dimensional of determinant $$\chi $$ χ , where $$\chi $$ χ is the mod p cyclotomic character. More precisely, we assume that $$\sigma _k$$ σ k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as $$k\rightarrow 2$$ k → 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to $$\rho $$ ρ ) which we assume are non-zero.

The explicit Sato–Tate conjecture for primes in arithmetic progressions

Let [Formula: see text] be Ramanujan’s tau function, defined by the discriminant modular form [Formula: see text] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that [Formula: see text] for all [Formula: see text]; since [Formula: see text] is multiplicative, it suffices to study primes [Formula: see text] for which [Formula: see text] might possibly be zero. Assuming standard conjectures for the twisted symmetric power [Formula: see text]-functions associated to [Formula: see text] (including GRH), we prove that if [Formula: see text], then [Formula: see text] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato–Tate conjecture for primes in arithmetic progressions.

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 $1$ .

Quantum modularity of partial theta series with periodic coefficients

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich–Zagier series F t ⁢ ( q ) \mathscr{F}_{t}(q) which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T ⁢ ( 3 , 2 t ) T(3,2^{t}) , t ≥ 2 t\geq 2 , is a weight 3 2 \frac{3}{2} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F ⁢ ( q ) F(q) .