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Jonas Bergström

Publications and source records attributed to Jonas Bergström.

29 records · Page 2Linked to original sources
arXiv2016

Eisenstein congruences for SO(4,3), SO(4,4), spinor and triple product L-values

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.

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arXiv2015

Eisenstein congruences for split reductive groups

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.

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arXiv2013

On the cohomology of moduli spaces of (weighted) stable rational curves

We give a recursive algorithm for computing the character of the cohomology of the moduli space $\Mmb_{0,n}$ of stable $n$-pointed genus zero curves as a representation of the symmetric group $\Sg_n$ on $n$ 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.

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arXiv2013

Siegel modular forms of degree three and the cohomology of local systems

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 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 G_2 and new congruences of Harder type.

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arXiv2011

Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves

We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for $\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results are applied to the moduli spaces of stable curves of genus 2 with up to 7 points, and this gives us the $\s_n$-equivariant Galois (resp. Hodge) structure of their $\ell$-adic (resp. Betti) cohomology.

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arXiv2009

Inversion of series and the cohomology of the moduli spaces $\mathcal{M}^δ_{0,n}$

For $n\geq 3$, let $\mathcal{M}_{0,n}$ denote the moduli space of genus 0 curves with $n$ marked points, and $\overline{\mathcal{M}}_{0,n}$ its smooth compactification. A theorem due to Ginzburg, Kapranov and Getzler states that the inverse of the exponential generating series for the Poincaré polynomial of $H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for $H^{\bullet}(\overline{\mathcal{M}}_{0,n})$. In this paper, we prove that the inverse of the ordinary generating series for the Poincaré polynomial of $H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for $H^{\bullet}(\mathcal{M}^δ_{0,n})$, where $\mathcal{M}_{0,n}\subset \mathcal{M}^δ_{0,n} \subset \overline{\mathcal{M}}_{0,n}$ is a certain smooth affine scheme.

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arXiv2008

Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the étale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine' Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures.

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arXiv2007

The rational cohomology of \Mbar_4

We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space \Mbar_4 of stable complex curves of genus 4, with its Hodge structure.

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arXiv2007

Cohomology of moduli spaces of curves of genus three via point counts

In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of genus 3 for $n \leq 5$ (to obtain $n \leq 7$ we would need counts of ``8-pointed curves of genus 2'').

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