For any square-free integer N such that the "moonshine group" Γ 0 (N ) + has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of Γ 0 (N ) + to certain McKay-Thompson series associated to the representation theory of the Fischer-Griess monster group. In particular, the Hauptmoduli admits a q-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus moonshine groups Γ 0 (N ) + . For all moonshine groups of genus up to and including three, we prove that the corresponding function field admits two generators whose q-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity. As corollary, we derive a polynomial relation which defines the underlying projective curve, and we deduce whether i∞ is a Weierstrass point. Our method of proof is based on modular forms and includes extensive computer assistance, which, at times, applied Gauss elimination to matrices with thousands of entries, each one of which was a rational number whose numerator and denominator were thousands of digits in length.

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups Γ 0 (N ) + , where N > 1 is a square-free integer. After we prove that Γ 0 (N ) + has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an "average" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the "classical" Weyl's law as a special case. The groups corresponding to N = 5 and N = 6 have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for Γ 0 (5) + than for Γ 0 (6) + . We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first 3557 of Γ 0 (5) + and the first 12474 eigenvalues of Γ 0 (6) + . With this information, we empirically verify some conjectured distributional properties of the eigenvalues.

No abstract

For automorphic L-functions L(s, π) and L(s, π ) attached to automorphic irreducible cuspidal representations π and π of GLm(QA) and GL m (QA), we prove the Selberg orthogonality unconditionally for m ≤ 4 and m ≤ 4, and under hypothesis H of Rudnik and Sarnak if m > 4 or m > 4, without the additional requirement that at least one of these representations be self-contragradient.

In order to determine the habitat preferred by Myodes (before Clethrionomys) glareolus and the corresponding Puumala hantavirus seroprevalence in those habitats, we captured rodents simultaneously in three significantly different habitats. We compared trapping success and presence of virus per habitat during an ongoing epidemic in order to test the hypothesis of a density-dependent seroprevalence. Our study showed that bank vole population density, as well as Puumala virus seroprevalence, were habitat dependent. Apodemus sylvaticus was found more vulnerable for deteriorating habitat conditions than M. glareolus and could play a role as vehicle for Puumala virus and as mediator for inter- and conspecific virus transmission.