Arithmetic geometry and automorphic forms

Invent Math, 1978

In this paper in the Section 1, we have described some equations concerning the duality and higher derivative terms in M-theory. In the Section 2, we have described some equations concerning the moduli-dependent coefficients of higher derivative interactions that appear in the low energy expansion of the four-supergraviton amplitude of maximally supersymmetric string theory compactified on a d-torus. Thence, some equations regarding the automorphic properties of low energy string amplitudes in various dimensions. In the Section 3, we have described some equations concerning the Eisenstein series for higher-rank groups, string theory amplitudes and string perturbation theory. In the Section 4, we have described some equations concerning U-duality invariant modular form for the D^6 R^4 interaction in the effective action of type IIB string theory compactified on T^2 . Furthermore, in the Section 5, we have described various possible mathematical connections between the arguments above mentioned and some sectors of Number Theory, principally the Aurea Ratio, some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan’s identities concerning π and the zeta strings. In conclusion, in the Appendix A, we have analyzed some pure numbers concerning various equations described in the present paper. Thence, we have obtained some useful mathematical connections with some sectors of Number Theory. In the Appendix B, we have showed the column “system” concerning the frequency system based on Phi and the table where we have showed the difference between the values of Phi^(n/7) and the values of the column “system” v1 26.02.2011 REVISITED VERSION 31.10.2020

2006

Journal of Number Theory, 2011

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both "naive" and "modified"), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.

Proceedings of the American Mathematical Society, 2012

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g ≥ 4, a new class of vectorvalued modular forms, defined on the Teichmüller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte χ 68 , which is a proof of the recently rediscovered Klein "amazing formula". Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, χ 68 and the theta series corresponding to the even unimodular lattices E 8 ⊕ E 8 and D + 16. We also find, for g = 4, a functional relation between the singular component of the theta divisor and the Riemann period matrix.

Commentarii Mathematici Helvetici, 1992

Journal of the American Mathematical Society, 1999

Let E / F E/F be a quadratic extension of number fields and G = Res E / F ⁡ H G= \operatorname {Res}_{E/F}H , where H H is a reductive group over F F . We define the integral (in general, non-convergent) of an automorphic form on G G over H ( F ) ∖ H ( A ) 1 H(F)\backslash H(\mathbb A)^1 via regularization. This regularized integral is used to derive a formula for the integral over H ( F ) ∖ H ( A ) 1 H(F)\backslash H(\mathbb A)^1 of a truncated Eisenstein series on G G . More explicit results are obtained in the case H = G L ( n ) H=GL(n) . These results will find applications in the expansion of the spectral side of the relative trace formula.

2016

Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482

2002

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1997

We start with a brief overview of the necessary theory: Given any cusp form f=∑ n≥ 1 an (f) qn of weight k, we denote by L (f, s) the L-function of f. For Re (s)> k/2+ 1, the value of L (f, s) is given by L (f, s)=∑ n≥ 1 an (f) ns and, one can show that L (f, s) has analytic continuation to the entire complex plane. The value of L (f, s) at s= k/2 will be of particular interest to us, and we will refer to this value as the central critical value of L (f, s). Let χD denote the Dirichlet character associated to the extension Q (√ D)/Q, that is χD (n)=(∆ D n), where∆ D ...