Exact standard zeta-values of Siegel modular forms

In this paper, we give exact values of the standard zeta function for cuspidal Hecke eigenforms with respect to Sp2(Z)

On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\bar{\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S) where $S$ is the Pontryagin dual of the Selmer group associated to $\rho_{f} \otimes \epsilon^{1-k}$ with $\epsilon$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.Comment: 33 page

Modular symbols in Iwasawa theory

This survey paper is focused on a connection between the geometry of $\mathrm{GL}_d$ and the arithmetic of $\mathrm{GL}_{d-1}$ over global fields, for integers $d \ge 2$. For $d = 2$ over $\mathbb{Q}$, there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for $d = 2$ over $\mathbb{F}_q(t)$. In the third, we pose questions for general $d$ over the rationals, imaginary quadratic fields, and global function fields.Comment: 43 page

Transcendental equations satisfied by the individual zeros of Riemann ζ\zeta, Dirichlet and modular LL-functions

We consider the non-trivial zeros of the Riemann $\zeta$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert $W$-function. Our approach is a novel and simple method, that takes into account $\arg L$, to numerically compute non-trivial zeros of $L$-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the $\zeta$-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet $L$-functions and the $L$-function for the modular form based on the Ramanujan $\tau$-function, which is closely related to the bosonic string partition function.Comment: Matches the version to appear in Communications in Number Theory and Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and arXiv:1307.8395 [math.NT

Higher Hida theory and p-adic L-functions for GSp(4)

We use the "higher Hida theory" recently introduced by the second author to p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4), contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree 4 (spin) L-function of automorphic representations of GSp(4), and the degree 8 L-function of GSp(4) x GL(2).Comment: Updated with minor corrections. To appear in "Duke Math Journal" (see https://projecteuclid.org/accepted/euclid.dmj