Time-line until the publicaton of Weil of "Numbers of solutions of equations in finite fields" In "On the history of the Weil Conjectures" Dieudonné says:
"Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...".


*

*C. F. Gauss, Disquisitiones Arithmeticae; Mathematisches Tagebuch.


*A letter to Gauss(1827), C. G. Jacobi.


*V. A. Lebesgue work on this type of equations(1837).


*Hardy and Littlewood work on the singular series for Waring's problem.

I know that: Hasse proved a conjecture due to Artin about the "Riemann Hypothesis" for curves of genus 1 over Finite fields, Artin conjectured the zeroes of a rational function with coefficients in Q would all be on the circle [u]=q^1/2, and verified that conjecture for many polynomials P of low degree, Weil proved the "Riemann Hypothesis" for a curve of arbitrary genus and for some kind of hypersurfaces and grassmannians among other types of varieties.
My questions are:

What happened between Hardy and Weil work?
Which are the papers of this works?
Is there any book that contains these works?

 A: Here are a few references that give some of the history between Hardy and Weil. 
Cartier, Pierre, Des nombres premiers à la géométrie algébrique (une brève histoire de la fonction zéta). (French) [From primes to algebraic geometry (a brief history of the zeta function)] Analyse diophantienne et géométrie algébrique, 51–77, Cahiers Sém. Hist. Math. Sér. 2, 3, Univ. Paris VI, Paris, 1993. MR1240754 (94i:11060) 
Houzel, Christian, La préhistoire des conjectures de Weil.  [The prehistory of the Weil conjectures] Development of mathematics 1900–1950 (Luxembourg, 1992), 385–414, Birkhäuser, Basel, 1994. MR1298637 (96g:14001) 
Rosen, Michael, Number theory in function fields.  Graduate Texts in Mathematics, 210. Springer-Verlag, New York, 2002. xii+358 pp. ISBN: 0-387-95335-3. MR1876657 (2003d:11171) 
Roquette, Peter, The Riemann hypothesis in characteristic p, its origin and development. I. The formation of the zeta-functions of Artin and of F. K. Schmidt. Hamburger Beiträge zur Geschichte der Mathematik. Mitt. Math. Ges. Hamburg 21 (2002), no. 2, 79–157.  MR1973592 (2004i:11136) 
Roquette, Peter, The Riemann hypothesis in characteristic p, its origin and development. Part 4. Davenport-Hasse fields. Mitt. Math. Ges. Hamburg 32 (2012), 145–210. MR3059379 
Freitag, Eberhard; Kiehl, Reinhardt, Étale cohomology and the Weil conjecture. 
Translated from the German by Betty S. Waterhouse and William C. Waterhouse. With an historical introduction by J. A. Dieudonné. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 13. Springer-Verlag, Berlin, 1988. xviii+317 pp. ISBN: 3-540-12175-7. MR0926276 (89f:14017) 
In the last reference, it's specifically the introduction by Dieudonné that has the historical information. 
A: The papers by Roquette
http://www.rzuser.uni-heidelberg.de/~ci3/rv.pdf, http://www.rzuser.uni-heidelberg.de/~ci3/rv2.pdf, http://www.rzuser.uni-heidelberg.de/~ci3/rv3.pdf
and
http://www.rzuser.uni-heidelberg.de/~ci3/rv4.pdf
seem to me to be very thorough on the transitional period.
