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The results of this paper can be divided into two parts, geometric andarithmetic. Let $X$ be a smooth projective curve over $\mathbb{C}$, and$e,\infty\in X(\mathbb{C})$ be distinct points. Let $L_n$ be the mixed Hodgestructure of functions on $\pi_1(X-\{\infty\},e)$ given by iterated integralsof length $\leq n$ (as defined by Hain). In the geometric part, inspired by awork of Darmon, Rotger, and Sols, we express the mixed Hodge extension$\mathbb{E}^\infty_{n,e}$ given by the weight filtration on$\frac{L_n}{L_{n-2}}$ in terms of certain null-homologous algebraic cycles on$X^{2n-1}$. As a corollary, we show that the extension$\mathbb{E}^\infty_{n,e}$ determines the point $\infty\in X-\{e\}$. Thearithmetic part of the paper gives some number-theoretic applications of thegeometric part. We assume that $X=X_0\otimes_K\mathbb{C}$ and $e,\infty\inX_0(K)$, where $K$ is a subfield of $\mathbb{C}$ and $X_0$ is a projectivecurve over $K$. Let $Jac$ be the Jacobian of $X_0$. We use the extension$\mathbb{E}^\infty_{n,e}$ to associate to each $Z\in CH_{n-1}(X_0^{2n-2})$ apoint $P_Z\in Jac(K)$, which can be described analytically in terms of iteratedintegrals. The proof of $K$-rationality of $P_Z$ uses that the algebraic cyclesconstructed in the geometric part of the paper are defined over $K$. Assuming acertain plausible hypothesis on the Hodge filtration on $L_n(X-\{\infty\},e)$holds, we show that an algebraic cycle $Z$ for which $P_Z$ is torsion, givesrise to relations between periods of $L_2(X-\{\infty\},e)$. Interestingly,these relations are non-trivial even when one takes $Z$ to be the diagonal of$X_0$. The geometric result of the paper in $n=2$ case, and the fact that onecan associate to $\mathbb{E}^\infty_{2,e}$ a family of points in $Jac(K)$, aredue to Darmon, Rotger, and Sols. Our contribution is in generalizing thepicture to higher weights.

Algebraic Cycles, Fundamental Group of a Punctured Curve, and Applications in Arithmetic