|dc.description.abstract||The first part of the dissertation is mainly from my first paper joint with my advisor Papanikolas and the second part will be our second paper.
In 1973, Serre introduced p-adic modular forms for a fixed prime p, which are defined to be p-adic limits of Fourier expansions of holomorphic modular forms on SL2(ℤ) with rational coefficients. He also established fundamental results about families of p-adic modular forms by developing the theories of differential operators and Hecke operators acting on p-adic spaces of modular forms. In particular, he showed that the weight 2 Eisenstein series E2 is also p-adic. If we let ϑ := 1 /2πi d/dz be Ramanujan’s theta operator acting on holomorphic complex forms, then letting q(z) = e^2πiz, we have ϑ = q d/dq; ϑ(q^n) = nq^n. Although ϑ does not preserve spaces of complex modular forms, Serre proved the induced operation ϑ : ℚ⨂ℤp[[q]] → ℚ⨂ℤp[[q]] does take p-adic modular forms to p-adic modular forms and preserves p-integrality. Moreover, the Bernoulli numbers Bm and the Eisenstein series Em have p-adic limits as m goes to a p-adic limit.
To extend the theory to function fields, we investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Similar to the classical case, the false Eisenstein series E is a v-adic modular form, though it is not a Drinfeld modular form. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality, which can be proved using Goss polynomials.
Furthermore, we can show that the Bernoulli-Carlitz numbers BCmj have a v-adic limit if mj have the form aq^dj + b with a, b non-negative. Using the same method, we can also prove that the Goss polynomials have v-adic limits after multiplication by a Carlitz factorial. Because of this, we can also prove the limit of ПmjΘ^mj exists. Therefore, since the Eisenstein series En can be expressed as the sum of Bernoulli-Carlitz numbers and Goss polynomials, we can derive that Emj also have a v-adic limit in K⨂AAv[[u]]. Notice for the Eisenstein series in function fields, the result we get is different from the classical number fields. In the classical case, Serre proved that if mj has a limit m in the p-adic topology and mj goes to infinity in the Euclidean norm, then the classical Eisenstein series Emj has a p-adic limit only depending onm. However, for example in function fields, even if the two series aq^dj + b and (q - 1)q^2dj + aq^dj + b satisfy the previous two condition and their corresponding Eisenstein series are non-zero, they do not have the same v-adic limit.||en