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.

On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

Flavor moonshine

The flavor moonshine hypothesis is formulated to suppose that all particle masses (leptons, quarks, Higgs, and gauge particles—more precisely, their mass ratios) are expressed as coefficients in the Fourier expansion of some modular forms just as, in mathematics, dimensions of representations of a certain group are expressed as coefficients in the Fourier expansion of some modular forms. The mysterious hierarchical structure of the quark and lepton masses is thus attributed to that of the Fourier coefficient matrices of certain modular forms. Our intention here is not to prove this hypothesis starting from some physical assumptions but rather to demonstrate that this hypothesis is experimentally verified and, assuming that the string theory correctly describes the natural law, to calculate the geometry (Kähler potential and the metric) of the moduli space of the Calabi–Yau manifold, thus providing a way to calculate the metric of the Calabi–Yau manifold itself directly from the experimental data.

On the coefficients of symmetric power L-functions

We study the signs of the Fourier coefficients of a newform. Let [Formula: see text] be a normalized newform of weight [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the [Formula: see text]th Fourier coefficient of [Formula: see text]. For any fixed positive integer [Formula: see text], we study the distribution of the signs of [Formula: see text], where [Formula: see text] runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power [Formula: see text]-functions.