A multi-variable version of the completed Riemann zeta function and other LL-functions

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure on its residues. The special case of two variables can be written as a partial Mellin transform of a real analytic Eisenstein series, which enables us to relate its values at pairs of positive even points to periods of (simple extensions of symmetric powers of the cohomology of) the CM elliptic curve corresponding to the Gaussian integers. In general, the totally even values of these functions are related to new quantities which we call multiple quadratic sums. More generally, we cautiously define multiple-variable versions of motivic $L$-functions and ask whether there is a relation between their special values and periods of general mixed motives. We show that all periods of mixed Tate motives over the integers, and all periods of motivic fundamental groups (or relative completions) of modular groups, are indeed special values of the multiple motivic $L$-values defined here.Comment: This is the second half of a talk given in honour of Ihara's 80th birthday, and will appear in the proceedings thereo

Some heuristics about elliptic curves

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about $c_1X^{19/24}(\log X)^{3/8}$ curves with $|\Delta|<X$ with even parity and positive (analytic) rank; since Brumer and McGuinness predict $cX^{5/6}$ total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions

Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem

Let $E_1$ and $E_2$ be $\overline{\mathbb{Q}}$-nonisogenous, semistable elliptic curves over $\mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by $a_{E_i}(p) := p+1-\#E_i(\mathbb{F}_p)$ the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power $L$-functions $L(s, \mathrm{Sym}^i E_1\otimes\mathrm{Sym}^j E_2)$ where $i,j\in\{0,1,2\}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p\nmid N_{E_1}N_{E_2}$ such that $a_{E_1}(p)a_{E_2}(p)<0$ and $$p < \big( (32+o(1))\cdot \log N_{E_1} N_{E_2}\big)^2.$$ This improves and makes explicit a result of Bucur and Kedlaya. Now, if $I\subset[-1,1]$ is a subinterval with Sato-Tate measure $\mu$ and if the symmetric power $L$-functions $L(s, \mathrm{Sym}^k E_1)$ are functorial and satisfy GRH for all $k \le 8/\mu$, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime $p\nmid N_{E_1}$ such that $a_{E_1}(p)/(2\sqrt{p})\in I$ and $$p < \left((21+o(1)) \cdot \mu^{-2}\log (N_{E_1}/\mu)\right)^2.$$Comment: 30 page

The effect of convolving families of L-functions on the underlying group symmetries

L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M --> oo, the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary, symplectic or orthogonal). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level density. (We also study convolved families of the form f x G_M for a fixed f.) Under natural assumptions on the families (which hold in many cases) we can associate to each family L of L-functions a symmetry constant c_L equal to 0 (resp., 1 or -1) if the corresponding low-lying zero statistics agree with those of the unitary (resp., symplectic or orthogonal) group. Our main result is that c_{F x G} = c_G * c_G: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f x G_M. We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N,M --> oo, as lower-order terms).Comment: 41 pages, version 2.1, cleaned up some of the text and weakened slightly some of the conditions in the main theorem, fixed a typ

Explicit lower bounds on the modular degree of an elliptic curve

We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result

Holonomy of the Ising model form factors

We study the Ising model two-point diagonal correlation function $C(N,N)$ by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable $\lambda$, the $j$-particle contributions, $f^{(j)}_{N,N}$. The corresponding $\lambda$ extension of the two-point diagonal correlation function, $C(N,N; \lambda)$, is shown, for arbitrary $\lambda$, to be a solution of the sigma form of the Painlev{\'e} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $f^{(j)}_{N,N}$ are obtained and shown to have both a ``Russian doll'' nesting, and a decomposition of the differential operators as a direct sum of operators equivalent to symmetric powers of the differential operator of the elliptic integral $E$. Each $f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $E$ and $K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll'' structure. The previous $\lambda$-extensions, $C(N,N; \lambda)$ are, for singled-out values $\lambda= \cos(\pi m/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painlev\'e VI are actually algebraic functions, being associated with modular curves.Comment: 39 page

The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra

We give a generators and relations presentation of the HOMFLYPT skein algebra $H$ of the torus $T^2$, and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that $H$ is isomorphic to the $t=q$ specialization of the elliptic Hall algebra of Burban and Schiffmann [BS12]. As an application, for an iterated cable $K$ of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the $\lambda$-colored Homflypt polynomial of $K$. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko [CD14] using the $\mathfrak{gl}_N$ double affine Hecke algebra. This proves one of the Connection Conjectures in [CD14].Comment: v1: preliminary version, 36 pages, many figures. v2: minor edits and improvements to expositio

A conditional determination of the average rank of elliptic curves

Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are exactly $\frac 12$. As a corollary we obtain that under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves in our family, and that asymptotically one half of these curves have algebraic rank $0$, and the remaining half $1$. We also prove an analogous result in the family of all elliptic curves. A way to interpret our results is to say that nonreal zeros of elliptic curve $L$-functions in a family have a direct influence on the average rank in this family. Results of Katz-Sarnak and of Young constitute a major ingredient in the proofs.Comment: 27 page