Browsing by Author "Hain, Richard M"
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Item Open Access Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves(2018) Luo, MaTo study periods of fundamental groups of algebraic varieties, one requires an explicit algebraic de Rham theory for completions of fundamental groups. This thesis develops such a theory in two cases. In the first case, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity. In the second case, we develop an algebriac de Rham theory for relative completion of the fundamental group of the moduli space of elliptic curves with one marked point. This allows the construction of iterated integrals involving modular forms of the second kind, whereas previously Brown and Manin only studied iterated integrals of holomorphic modular forms.
Item Open Access Cohomology Jumping Loci and the Relative Malcev Completion(2007-12-12) Narkawicz, Anthony JosephTwo standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev completion of its fundamental group G and the cohomology groups of X with coefficients in rank one local systems. In this thesis, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. The relative completion S_p is a prosolvable group that generalizes the classical Malcev completion; when p is the trivial representation, S_p is the Malcev completion of G. The group S_p is tightly controlled by the cohomology groups H^1(X,L_{p^k}) with coefficients in the irreducible local systems L_{p^k} associated to the representation p.The pronilpotent Lie algebra u_p of the prounipotent radical U_p of S_p has been described by Hain. If p is the trivial representation, then u_p is the holonomy Lie algebra, which is well-known to be quadratically presented. In contrast, we show that when X is the complement of the braid arrangement in complex two-space, there are infinitely many representations p from G into (C^*)^2 for which u_p is not quadratically presented.We show that if Y is a subtorus of the character torus T containing the trivial character, then S_p is combinatorially determined for general p in Y. We do not know whether S_p is always combinatorially determined. If S_p is combinatorially determined for all characters p of G, then the characteristic varieties of the arrangement X are combinatorially determined.When Y is an irreducible subvariety of T^N, we examine the behavior of S_p as p varies in Y. We define an affine group scheme S_Y over Y such that if Y = {p}, then S_Y is the relative Malcev completion S_p. For each p in Y, there is a canonical homomorphism of affine group schemes from S_p into the affine group scheme which is the restriction of S_Y to p. This is often an isomorphism. For example, if there exists p in Y whose image is Zariski dense in G_m^N, then this homomorphism is an isomorphism for general p in Y.Item Open Access Cyclotomic and Elliptic Polylogarithms and Motivic Extensions of Q by Q(m)(2021) Hopper, Eric JeffreyDeligne and Goncharov constructed a neutral tannakian category of mixed Tate motives MTM_N unramified over Z[mu_N,1/N]. Brown and Hain--Matsumoto computed the first order action of its motivic Galois group on the unipotent fundamental group of a smoothing of the once punctured Tate elliptic curve when N = 1. In this thesis, we extend their work to N > 1 by computing the first order action of the motivic Galois group on the unipotent fundamental group of a smoothing of the Tate elliptic curve with a cyclic subgroup of order N removed.
One of our main tools is the elliptic KZB connection for the level N congruence subgroup of SL_2(Z). We adapt it to the arithmetic setting and prove that it underlies an admissible variation of mixed Hodge structure over the corresponding universal elliptic curve with its N-torsion removed. At the singular fiber above the distinguished cusp q = 0, we show that the KZB connection degenerates to the cyclotomic KZ connection and that the variation degenerates to a mixed Hodge structure that contains the Lie algebra of the motivic fundamental group pi_1^mot(P^1 - {0,mu_N,infty},v) studied by Deligne and Goncharov. This observation allows us to construct a higher level analogue of the Hain map. We further prove this map is a morphism in MTM_N by describing the Galois action on the motivic periods of the unipotent fundamental group of the punctured Tate curve.
The inclusion of pi_1^mot(P^1 - {0,mu_N,infty},v) into the limit mixed Hodge structure of the KZB variation allows us to relate the periods of the N-cyclotomic polylog variation of MHS to the periods of the level N elliptic polylog variation of MHS. This enables us to give a formula for the first order action of the Galois group of MTM_N on the elliptic polylogarithm. This is most explicit when N is a prime power.
Item Open Access Hodge Theory and String Topology(2021) Xiong, XinLet $M$ be an oriented smooth manifold of dimension $n$. The free loop space $\Lambda M$ of $M$ is the space of piecewise smooth maps from the circle to $M$. Chas and Sullivan defined the string product$$ CS : H_i (\Lambda M;\\mathbb{Z}) \otimes H_j (\Lambda M; \mathbb{Z}) \to H_{i + j - n} (\Lambda M; \mathbb{Z}). $$ Goresky and Hingston defined a string coproduct $$ GH: H_k (\Lambda M, M; \mathbb{Q}) \to \bigoplus_{i + j = k - n + 1} H_i (\Lambda M, M; \mathbb{Q}) \otimes H_j (\Lambda M, M; \mathbb{Q}). $$ When $M$ is a simply-connected complex algebraic manifold, the cohomology of $\Lambda M$ has a natural mixed Hodge structure with weights greater or equal to degree. In this case, we show that the string operations, $CS$ and $GH$, are morphisms of mixed Hodge structure after a suitable Tate twist. We prove this by giving a de Rham description of the string operations in terms of Chen's iterated integrals.
Item Open Access Rational Points of Universal Curves in Positive Characteristics(2015) Watanabe, TatsunariFor the moduli stack $\mathcal{M}_{g,n/\mathbb{F}_p}$ of smooth curves of type $(g,n)$ over Spec $\mathbb{F}_p$ with the function field $K$, we show that if $g\geq3$, then the only $K$-rational points of the generic curve over $K$ are its $n$ tautological points. Furthermore, we show that if $g\geq 3$ and $n=0$, then Grothendieck's Section Conjecture holds for the generic curve over $K$. A primary tool used in this thesis is the theory of weighted completion developed by Richard Hain and Makoto Matsumoto.
Item Open Access Theta Functions and the Structure of Torelli Groups in Low Genus(2015) Kordek, Kevin AThe Torelli group Tg of a closed orientable surface Sg of genus g >1 is the group
of isotopy classes of orientation-preserving diffeomorphisms of Sg which act trivially
on its first integral homology. The hyperelliptic Torelli group TDg is the subgroup
of Tg whose elements commute with a fixed hyperelliptic involution. The finiteness
properties of Tg and TDg are not well-understood when g > 2. In particular, it is not
known if T3 is finitely presented or if TD3 is finitely generated. In this thesis, we begin
a study of the finiteness properties of genus 3 Torelli groups using techniques from
complex analytic geometry. The Torelli space T3 is the moduli space of non-singular
genus 3 curves equipped with a symplectic basis for the first integral homology and is
a model of the classifying space of T. Each component of the hyperelliptic locus T hyp 3
in T3 is a model of the classifying space for TD3. We will investigate the topology
of the zero loci of certain theta functions and thetanulls and explain how these are
related to the topology of T3 and T3 hyp. We show that the zero locus in h 2 x C2
of any genus 2 theta function is isomorphic to the universal cover of the universal framed genus 2 curve of compact type and that it is homotopy equivalent to an infinite bouquet of 2-spheres. We also derive a necessary and sufficient condition for the zero locus of any genus 3 even thetanull to be homotopy equivalent to a bouquet of 2-spheres and 3-spheres.
Item Open Access Triple Products of Eisenstein Series(2015) Venkatesh, AnilIn this thesis, we construct a Massey triple product on the Deligne cohomology of the modular curve with coefficients in symmetric powers of the standard representation of the modular group. This result is obtained by constructing a Massey triple product on the extension groups in the category of admissible variations of mixed Hodge structure over the modular curve, which induces the desired construction on Deligne cohomology. The result extends Brown's construction of the cup product on Deligne cohomology to a higher cohomological product.
Massey triple products on Deligne cohomology have been previously investigated by Deninger, who considered Deligne cohomology with trivial real coefficients. By working over the reals, Deninger was able to compute cohomology exclusively with differential forms. In this work, Deligne cohomology is studied over the rationals, which introduces an obstruction to applying Deninger's results. The obstruction arises from the fact that the integration map from the de Rham complex to the Eilenberg-MacLane complex of the modular group is not an algebra homomorphism. We compute the correction terms of the integration map as regularized iterated integrals of Eisenstein series, and show that these integrals arise in the cup product and Massey triple product on Deligne cohomology.