Let q be a power of a prime number p, F-q be a finite field with q elements and G be a subgroup of (F-q,+) of order p. We give an existence criterion and an algorithm for computing maximally G-fixed c-Wieferich primes in F-q[T]. Using the criterion, we study how c-Wieferich primes behave in F-q[T] extensions.
We express the coefficients of the Hirzebruch L-polynomials in terms of certain alternating multiple zeta values. In particular, we show that every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign. Similar results hold for the polynomials associated to the Â-genus.
We compute the weighted Euler characteristic, equivariant with respect to the action of the symplectic group of degree six over the field of two elements, of the moduli space of principally polarized Abelian threefolds together with a level two structure.
We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain L-functions. We examine the consequences in several special cases and use the Bloch–Kato conjecture to further motivate a belief in the congruences.
We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as GSp2(A), SO(4,3)(A) and SO(5,4)(A), where the prime modulus should, for various reasons, appear in the algebraic part of a critical “tensor-product” L-value associated to cuspidal automorphic representations of GL2(A) and GSp2(A). Using special techniques for evaluating L-functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors.
We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a split orthogonal group. We provide some numerical evidence in the case that the group is SO(4, 3) and the L-function is the spinor L-function of a genus 2, vector-valued, Siegel cusp form. We also consider the case that the group is SO(4, 4) and the L-function is a triple product L-function.
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of degree three, such as dimension formulas and traces of Hecke operators. We also use it to predict several lifts from genus 1 to genus 3, as well as lifts from and new congruences of Harder type.
We give a recursive algorithm for computing the character of the cohomology of the moduli space of stable -pointed genus zero curves as a representation of the symmetric group on letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett.
We determine the cohomology of the Losev--Manin moduli space $\overline{M}_{0, 2 | n}$ of pointed genus zero curves as a representation of the product of symmetric groups $\Sg_2 \times \Sg_n$.
We formulate a detailed conjectural Eichler–Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes GU(2, 1, Q (√-3)). The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula.
Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems.
We provide explicit generators of the torsion of the second cohomology of bielliptic surfaces, and we use this to study the pullback map between the Brauer group of a bielliptic surface and that of its canonical cover.