The mixed Hodge structure on the fundamental group of a punctured Riemann surface

Abstract

Abstract. Given a compact Riemann surfaceX of genus g and distinct points p and q onX, we consider the non-compact Riemann surface X :=X \ {q} with basepoint p ∈ X. The extension of mixed Hodge structures associated to the first two steps of π 1 (X, p) is studied. We show that it determines the element (2g q − 2 p − K) in Pic 0 (X), where K represents the canonical divisor ofX as well as the corresponding extension associated to π 1 (X, p). Finally, we deduce a pointed Torelli theorem for punctured Riemann surfaces.

Quadratic Chabauty and rational points, I: p-adic heights

“…In [19, Theorem 2.2], Darmon, Rotger and Sols proved that the Abel-Jacobi class of D(b, z) is equal to the extension of Z-mixed Hodge structure corresponding to the motive whose étale realisation is IA(b, z). This generalised previous work of Kaenders [29]. The theorem below refines this to determine A(b, z) as a mixed extension of κ(z − b) and IA(b, z) * (1).…”

“…To this end, we run the Mordell-Weil sieve (modified as above) with S containing primes above 7, 13,17,29,101,109,199,239,313,373,677,757. We finally show that no odd prime divides both lcm ({#J(k v ) : v ∈ S}) and (J(K) : G); this proves that we indeed have X(K) = {(±2 : ±1), (±i, ±4), ∞ ± }, thus finishing the proof of Theorem 8.8.…”

The fundamental group and extensions of motives of Jacobians of curves

“…obtained by restricting E 3 Q,P to the extension of S by H 1 (C). From Kaenders [Kae01] one knows there is a covering map of complex tori,…”

“…It is well known that Ext(S, H 1 (C)) = Ext(Z(−1), H 1 (C)) ≃ Pic 0 (C). To understand the other term, from the work of Hain [Hai87], Pulte [Pul88], Kaenders [Kae01] and Rabi [Rab01] one has the following theorem…”

The mixed Hodge structure on the fundamental group of hyperelliptic curves and higher cycles

“…The theory of iterated integrals for pointed Riemann surfaces (C, p) and pointed punctured ones (C − {q}, p) describes explicitely the canonical MHS on the quotients J p /J k p and J q,p /J k q,p , where J p := ker(ǫ : Zπ 1 (C, p) → Z) and J q,p := ker(ǫ : Zπ 1 (C − {q}, p) → Z) (we refer to [10] and in particular for punctured curves to [17]). The weight filtrations on the duals are given, for l ≤ k, by W l (J p /J k p ) * := (J p /J l+1 p ) * and W l (J q,p /J k q,p ) * := (J q,p /J l+1 q,p ) * .…”