*Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.*

Thursday, May 5, 2016

11:10 a.m.

BA6183

PhD Candidate: Payman Eskandari

Supervisor: Kumar Murty

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

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Abstract:

The results of this thesis can be divided into two parts, geometric and arithmetic. 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 Hodge structure of functions on $\pi_1(X-\{\infty\},e)$ given by iterated integrals of length $\leq n$ (as defined by Hain). In the geometric part, inspired by a work of Darmon, Rotger, and Sols, we express the mixed Hodge extension $\mathbb{E}^\infty_{n,e}$ given by the weight filtration on $\displaystyle{\frac{L_n}{L_{n-2}}}$ in terms of certain null-homologous algebraic cycles on $X^{2n-1}$. These cycles are constructed using the diagonal embeddings of $X^{n-1}$ into $X^n$.

The arithmetic part of the thesis gives some number-theoretic applications of the geometric part. We assume that $X=X_0\otimes_K\mathbb{C}$ and $e,\infty\in X_0(K)$, where $K$ is a subfield of $\mathbb{C}$ and $X_0$ is a projective curve over $K$. Let $Jac$ be the Jacobian of $X_0$. We use the extension $\mathbb{E}^\infty_{n,e}$ to associate to each $Z\in CH_{n-1}(X_0^{2n-2})$ a point $P_Z\in Jac(K)$, which can be described analytically in terms of iterated integrals. The proof of $K$-rationality of $P_Z$ uses that the algebraic cycles constructed in the first part are defined over $K$. Assuming a certain plausible hypothesis on the Hodge filtration on $L_n$ holds, we show that an algebraic cycle $Z\in CH_{n-1}(X_0^{2n-2})$ for which $P_Z$ is torsion, gives rise to relations between periods of $L_2$. Interestingly, these relations are non-trivial even when one takes $Z$ to be the diagonal in $X_0^2$. In the elliptic curve case, we show unconditionally that a certain relation between periods of $L_2$ (which is induced by the diagonal in $X_0^2$) exists if and only if $e-\infty$ is torsion.

The geometric result of the thesis in $n=2$ case, and the fact that one can associate to $\mathbb{E}^\infty_{2,e}$ a family of points in $Jac(K)$, are due to Darmon, Rotger, and Sols. Our contribution is in generalizing the picture to higher weights.

A copy of the thesis can be found here: Eskandari_thesis

Exam PhD