We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules defined over the polynomial ring F_q[theta]. This generalizes results of Anderson for the rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.
Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden.
Introduced by Anglès, Pellarin, and Tavares Ribeiro, Drinfeld modules over Tate algebras are closely connected to Anderson log-algebraicity identities, Pellarin $L$-series, and Taelman class modules. In the present paper we define the de Rham map for Drinfeld modules over Tate algebras, and we prove that it is an isomorphism under natural hypotheses. As part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of Drinfeld modules over Tate algebras.
This is the second of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present part, we compare the analytic theory with the algebraic one that was begun in a paper of the third author. For any arithmetic congruence subgroup and any integral weight we establish an isomorphism between the space of analytic modular forms with the space of algebraic modular forms defined in terms of the Satake compactification. From this we deduce the important result that this space is finite dimensional.
We propose a lower bound estimate in Dobrowolski's style of the canonical height on a certain family of Drinfeld modules of characteristic 0, including under some hypothesis on their degree and their base field, the complex multiplication case, extending so a previous result of L. Denis on Carlitz modules. Our study is focused on the highest possible level of precision on the parameters involved with rapport to the main values which characterize the Drinfeld module (height, base field degree and rank) and it provides an estimate in function of both separable and inseparable degree of the algeb…
证明了符号归一化秩为1的Drinfeld模所对应的Stark单位的模可以由Anderson等变$A$ -调和级数得到.我们应用这一点来获得一个类公式à la Taelman和证明一个多变量的对数代数性定理，推广安德森的对数代数性定理。我们还给出了Anderson的对数代数性定理的另一个证明，并得到了关于任意秩Drinfeld模的Stark单位的模的各种结果。
In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number field $K$. In this paper we establish Riesz sum identities containing an "extra" parameter and involving $f_K(n)$, or divisor f…
Based on the notion of Stark units we present a new approach that obtains refinements of log-algebraic identities for Anderson t-modules. As a consequence, we establish a generalization of Chang's theorem on logarithmic interpretations for special characteristic p multiple zeta values (MZV's) and recover many earlier results in this direction. Further, we devise a direct and conceptual way to get logarithmic interpretations for both MZV's and v-adic MZV's. This generalizes completely the work of Anderson and Thakur for Carlitz zeta values.
We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical $F$-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This $F$-…
We study algebraic cycles in the moduli space of $\mathrm{PGL}_2$-shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner-Drinfeld cycle is the product of the $r$-th central derivative of an automorphic $L$-function $L(π,s)$ and Waldspurger's toric period integral. When $L(π,\frac12) \neq 0$, this gives a new geometric interpretation for the Taylor series expansion. When $L(π,\frac12) = 0$, the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof …
We analyze log-algebraic power series identities for formal groups of elliptic curves over $\mathbb{Q}$ which arise from modular parametrizations. We further investigate applications to special values of elliptic curve $L$-functions.
For applications in algebraic geometric codes, an explicit description of bases of Riemann-Roch spaces of divisors on function fields over finite fields is needed. We investigate the third function field $ F^{(3)} $ in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vl{ă}du{ţ} bound. We construct bases for the related Riemann-Roch spaces on $ F^{(3)} $ and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on $ F^{(3)} $. All of these results can b…
Algebraic geometric framework of Rogers--Ramanujan identities
Emory University（埃默里大学）；Axiom Math（Axiom 数学研究所）；Research Institute for Symbolic Computation (RISC), Johannes Kepler University（约翰内斯·开普勒大学符号计算研究所（RISC））；Center for Applied Mathematics (TCAM), Tianjin University（天津大学应用数学中心（TCAM））
