Related papers: A note on higher Green's functions

Harer-Zagier generating functions for Euler characteristics of moduli spaces of curves contain $n$-necklace polynomials. Taylor expansions for these polynomials depend on numbers of solutions of Cohen semilinear congruences.

We show that the higher Chow groups with modulus of Binda-Kerz-Saito for a smooth quasi-projective scheme $X$ is a module over the Chow ring of $X$. From this, we deduce certain pull-backs, the projective bundle formula, and the blow-up…

We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each…

We prove a moving lemma for higher Chow groups with modulus, in the sense of Binda-Kerz-Saito, of projective schemes when the modulus is given by a very ample divisor. This provides one of the first cases of moving lemmas for cycles with…

The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.

We prove the Geometric Syzygy Conjecture for generic canonical curves of even genus. This result extends Green's classical result on the generation of the ideal of a canonical curve by rank four quadrics to the highest linear syzygy group.

Zagier observed that modular Nahm sums associated with the same matrix may form a vector-valued modular function on some congruence subgroup. We establish modular transformation formulas for several families of Nahm sums by viewing them as…

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers…

In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green's function of an asymptotically flat $3$-manifold. In the same context and for…

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures…

We study universal families of stable genus two curves with level structure. Among other things, it is shown that the (1,1) part is spanned by divisor classes, and that there are no cycles of type (2,2) in the third cohomology of the first…

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series $\mathscr{F}_t(q)$ which matches (at a root of unity) the colored Jones…

A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been explained. Various classical examples of this theorem, such as the Green's and…

We prove the existence of a module for the largest Mathieu group, whose trace functions are weight two quasimodular forms. Restricting to the subgroup fixing a point, we see that the integrality of these functions is equivalent to certain…

The theorems of Gross-Zagier and Zhang relate the N\'eron-Tate heights of complex multiplication points on the modular curve X_0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-functions. We extend these…

Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a…

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In…

Let E be an elliptic curve over Q, and let F be a finite abelian extension of Q. Using Beilinson's theorem on a suitable modular curve, we prove a weak version of Zagier's conjecture for L(E/F,2), where E/F is the base extension of E to F.

Recent work on the quantization of Maxwell theory has used a non-covariant class of gauge-averaging functionals which include explicitly the effects of the extrinsic-curvature tensor of the boundary, or covariant gauges which, unlike the…

We survey three results on syzygies of curves beyond Green's conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory.