## View abstract

### Session S21 - Galois representations and automorphic forms

Thursday, July 15, 15:20 ~ 16:00 UTC-3

## Overconvergent Eichler–Shimura morphism for families of Siegel modular forms

### Giovanni Rosso

#### Concordia University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak3e668d118e186100e0389ded7400ba89').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy3e668d118e186100e0389ded7400ba89 = 'g&#105;&#111;v&#97;nn&#105;.r&#111;ss&#111;' + '&#64;'; addy3e668d118e186100e0389ded7400ba89 = addy3e668d118e186100e0389ded7400ba89 + 'c&#111;nc&#111;rd&#105;&#97;' + '&#46;' + 'c&#97;'; var addy_text3e668d118e186100e0389ded7400ba89 = 'g&#105;&#111;v&#97;nn&#105;.r&#111;ss&#111;' + '&#64;' + 'c&#111;nc&#111;rd&#105;&#97;' + '&#46;' + 'c&#97;';document.getElementById('cloak3e668d118e186100e0389ded7400ba89').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy3e668d118e186100e0389ded7400ba89 + '\'>'+addy_text3e668d118e186100e0389ded7400ba89+'<\/a>';

Classical results of Eichler and Shimura decompose the cohomology of certain local systems on the modular curve in terms of holomorphic and anti-holomorphic modular forms. A similar result has been proved by Faltings' for the étale cohomology of the modular curve and Falting's result has been partly generalised to Coleman families by Andreatta–Iovita–Stevens. In this talk, based on joint work with Hansheng Diao and Ju-Feng Wu, I will explain how one constructs a morphism from the overconvergent cohomology of $\mathrm{GSp}_{2g}$ to the space of families of Siegel modular forms. This can be seen as a first step in an Eichler–Shimura decomposition for overconvergent cohomology and involves a new definition of the sheaf of overconvergent Siegel modular forms using the Hodge–Tate map at infinite level.

Joint work with Hansheng Diao (Tsinghua University, Yau Mathematical Sciences Center, China) and Ju-Feng Wu (Concordia University, Canada).