Seminar : Hilbert modular forms and the Gross-Stark conjecture

In this seminar we study the article [DDP]. Gross's conjecture says that for $F$ a totally real field, all characters $\chi$ of $F$ and $S = R \cup \{ \mathfrak{p} \}$, $p$ dividing $\mathfrak{p}$: $$ L'_{S,p}(\chi \omega, 0) = {\cal{L}}(\chi) L_R(\chi,0) $$ The paper confirms this conjecture in some cases, one for instance when Leopoldt's conjecture holds for $F$ and there are at least two primes above $p$ in $F$.

Literature

[DDP] Darmon, Dasgupta, Pollack, Hilbert modular forms and the Gross-Stark conjecture, 2011
[Hid93]
[Ser73] Serre, Formes modulaires et fonctions zeta $p$-adiques

Meetings

Friday, June 28, 2013
14:00-16:00	V5-227	Andreas	Weight one localization and Galois representations, II [Abstract] We prove Gross's Conjecture under the conditions stated in Theorem 2. More precisely, we establish that the analytic $L$-invariant is equal to the $L$-invariant by considering the cohomological interpretation of the conjecture. We construct a cohomology class $\kappa \in H^1_{\mathfrak{p}}(F,E(\chi^{-1}))$ that involves the special eigenvalues of $H_{1+\epsilon}$ or what is the same $F_{1+\epsilon}$, in particular the analytic $L$-invariant pops up.
Friday, June 19, 2013
14:00-16:00	V5-227	Andreas	Weight one localization and Galois representations, I [Abstract] We follow Section 3 and 4 of [DDP]. We use the normalized cuspidal eigenform $F_{1+\epsilon}$ having the same system of Hecke eigenvalues as $H_{1+\epsilon}$ to obtain a big Galois representation $\rho$. The coefficient ring is this time a connected component of the total ring of fractions of the localized Hecke ring $\mathbb{T}_(1)$.
Friday, June 14, 2013
14:00-15:30	V4-116	Hendrik	Big Galois representations, II [Abstract] We prove another patching theorem and show how to obtain by a simple trick a representation from a pseudo representation. The existence of $\rho_F$, where $F$ is an $I$-adic eigenform, is shown and all wished-for properties come directly from the corresponding properties of Deligne representations. Finally we show that if $F$ is ordinary, then the restriction of $\rho_F$ to a decomposition group at $p$ looks like $$ \left(\begin{array}{cc}\chi \cdot \kappa \cdot \chi_{\text{cyc}} & * \\0 & \lambda(a_p(F)) \end{array}\right) $$ where $\chi$ is the nebentypus of $F$.
Friday, June 14, 2013
14:00-15:30	V4-116	Hendrik	Big Galois representations, I [Abstract] We construct big Galois representations of $I$-adic eigen forms. First we recalled the classical results to attach a Galois representation to a classical eigenform. The notion of ordinary was also recalled. We stated the main theorem that a big galois representation $\rho_F$ attached to an $I$-adic eigenform $F$ exists. We defined (odd, $2$-dimensional) pseudo representations and proved a first patching lemma.
Friday, June 7, 2013
14:00-15:30	V3-200	Felix	Construction of $p$-adic L-series [Abstract] Following [Ser73], $p$-adic modular forms are defined as limits of classical modular forms. Only the case that the classical modular forms have level $1$ and are defined over $\mathbb{Q}$ is considered. We refind the $p$-adic $L$-series in the constant term of the $p$-adic Eisenstein series.
Tuesday, May 28, 2013
14:00-15:30	V5-227	Lennart	Global Galois Cohomology [Abstract] We continue the discussion of Galois cohomology, now focusing on global cohomology and cover \S 1.2 of the article. We prove that the connecting homomorphism $$ \delta : U_{\chi} \rightarrow H^1_{\mathfrak{p}}(F,E(1)(\chi)) $$ from the Kummer sequence is an isomorphism by constructing explicitly an inverse. The fact that this is an isomorphism, together with a dimension calculation of the space $H^1_{\mathfrak{p},\text{cyc}}(F,E(\chi^{-1}))$ (it's of dimension one) allows us to give a formula of the $L$-invariant of $\chi$ in terms of two cohomology classes. Hence we reduced proving the conjecture in computing explicitly such classes.
Tuesday, May 21, 2013
14:00-15:30	V5-227	Lennart	Local Galois Cohomology [Abstract] We defined unramified cohomology $H^_{nr}(K,V)$, where $V$ is a $p$-adic vector space and $K$ an $\ell$-adic field. Local Tate-duality for $H^1$ was made explicit by using the Kummer sequence. We stated the Euler-Poincare identity for $\ell \not = p$ as $$ \sum (-1)^i \dim H^i(K,V) = 0 $$ and under the Tate pairing, $H^1_{nr}(K,V^(1))$ is the orthogonal complement of $H^1_{nr}(K,V)$. Also an Euler-Poincare identity was stated for $\ell=p$ as $$ \sum (-1)^i \dim H^i(K,V) = -[K:\mathbb{Q}_p] \cdot \dim V . $$ Finally we defined $H^1_f(K,V)$.
Friday, April 26, 2013
14:00-15:30	V5-227	Michael	Introduction, part 3 [Abstract] The Hecke algebra for $\Lambda$-adic modular forms is briefly discussed and Hida's ordinary projector is introduced. The main point is to define the $\Lambda$-adic form $H$, a linear combination of two Eisenstein forms together with a certain $\Lambda$-adic form $P$. The talk basically covers the material of \S 2.3, 2.4 and 2.5 of the article.
Friday, April 26, 2013
14:00-15:30	V5-227	Michael	Introduction, part 2 [Abstract] A cohomological description of the Gross regulator is given. A $\Lambda$-adic form is defined, and the $\Lambda$-adic Eisenstein series are given as an example. The definition of the infinitesimal specialization map $\nu_{1+\epsilon}$ is given.
Friday, April 26, 2013
14:00-15:30	V5-227	Michael	Introduction, part 1 [Abstract] A general introductionary overview is given of the paper.

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Seminar : Hilbert modular forms and the Gross-Stark conjecture

Literature

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Friday, June 28, 2013

Friday, June 19, 2013

Friday, June 14, 2013

Friday, June 14, 2013

Friday, June 7, 2013

Tuesday, May 28, 2013

Tuesday, May 21, 2013

Friday, April 26, 2013

Friday, April 26, 2013

Friday, April 26, 2013