Using the theory of algebraic geometry and theta functions, we discovered a surprisingly simple formula for the dimension of the space of cusp forms of weight one associated to the Fermat curves. We also gave a basis for the concerned space. We notice such a formula does not exist for congruence groups.
Using the theory of theta lifting, we discovered a formula to the express central L-value of certain Hecke characters of a CM number field as the inner product of some theta lifting from $ U(1)$ to itself. It follows immediately that the central L-value is nonnegative which is also predicted by the generalized Riemann hypothesis. The formula also recovers a theorem of Rogawski on nonvanishing of global theta lifting in a natural way. Other applications are given in papers 4-6.
We constructed {\it{explicit}} eigenfunctions of Weil representation of unitary groups of one variable by means of lattice model. In addition to its own interest, it is also needed in the next three papers.
By computing the theta lifting in the formula of Paper 1, we discovered an interesting formula to express central Hecke L-values of CM number fields in terms of special values of classical theta functions at CM points. The theta functions involved is {\it{only}} dependent on the totally real subfield (not on the CM number field itself). In particular, we proved that the central L-value $L(k+1, (\chi_{p, d})^{2k+1})=0$ if and only if all the Heegner points of $X_0(4 d^2)$ with endomorphism ring $\Cal O_E$ are roots of a theta function only dependent on $d$ and $k$. Here $E=\Bbb Q(\sqrt{-p})$, $p\equiv 7 \mod 8$, $d \equiv 1 \mod 4$, every prime divisor of $d$ is split in $\Bbb Q(\sqrt{-p})$, $(-1)^k =\hbox{sign} (d)$, and $(2k+1, h_p) =1$. Also $\chi_{p, d}$ is a Hecke character of $E$ associated to the elliptic curve $A(p)^d$ defined by Gross in his thesis (LNM 776). Applying this result and a little calculus, we proved that for every integer $k \ge 0$, there is an integer $M(k)$ such that for all the pairs $(p,d)$ satisfying the conditions just mentioned and $\sqrt p > M(k) d^2$, the central L-value $L(k+1, (\chi_{p, d})^{2k+1})\ne 0$. We also computed $M(0)$ and $M(1)$ explicitly.
In this paper, we tried to remove the condition that every prime divisor of $d$ is split in $\Bbb Q(\sqrt{-p})$ from the result mentioned in the above paper. We succeeded the goal by overcoming three techinal problems. One result in this paper is as follows: The elliptic curve $A(p)^d$ defined in Gross's thesis (LNM 776) has rank $0$ if $p\equiv 7 \mod 8$, $d \equiv 1 \mod 4$, and $\sqrt p > d^2 \log d$. Moreover When every prime factor of $d$ is inert in $\Bbb Q(\sqrt{-p})$, and is congruent to 1 modulo 4, then $d^2$ can be replaced by $d$. $ (d >0)$.
In this note, we gave two applications of the main formula in paper 4 when the totally real field is a real quadratic number field.
In this note, we computed the special value of the L-series of the genus two curve in the title at the center. We are trying to study its arithmetic implications.
The general philosophy is that if a `nice' function from number theory is forced to be zero at its center, its deirvative at the center should be very interesting and might be linked to arithmetic of a geometric subject. The most famous example is the Gross-Zagier formula. Recent work of Kudla on the central derivative of an Eisenstein series is another example. There is a symtematic way to produce Eisenstein series on $Sp(n)$ which vanishes at the center from `incompatible' local quadratic space system of dimension $n+1$. Kudla has a general scheme to compute the central derivative of such Eisenstein series, and conjecture that its restriction on subgroups of $Sp(n)$ (by doubling method for example) are closely related to the global intesection numbers of cycles on Shimura varieties arising naturally. Moreover, this identity should be able to be proved term by term at each prime. He verified the case $n=2$ at good primes and infinity. As usual, bad primes cause a lot of technical difficulty. This joint paper is an attempt to give a complete picture in a very simple example, i.e., Eisenstein series on $Sp(1)$ coming from `incompatible' local quadratic space system of dimension $2$. We succeeded and the resulting central derivative of this Eisenstein series turns out be a very interesting non-holmorphic modular form of weight 1. Its Mellon transform looks particularly neat. Moreover, we give a geometric interpretation of the Fourier coefficients of the resulting modular form. We also learned some tricks to attack in general situaions.
This is both a continuation of project B and an example to realize the general philosophy just mentioned. Let $\chi$ be a Hecke character studied in project B. When its global root number is one, it was studied in project B. When its global root number is $-1$, the central L-value is automatically zero, and it is very interesting and natural to study its central derivative. We have written the L-function as the integral of certain {\bf{explicit}} Eisenstein series using doubling method. We have also computed the central derivative of the Eisenstein series using Kudla's scheme, including bad primes. We further computed its integral, and thus obtained a formula for the central derivative of the Hecke L-function. There are still two problems to work with. One is to make sense the formula, i.e., to give some kind of geometric interpretation. The other is to somwhow further simplify the formula and to derive some nonvanishing result similar to that of project B. This would in particular imply that Gross's elliptic curves $A(p)^d$ has rational rank one under certain easy to verify conditions. In this note, we found an explicit formula for the central derivative L'(1, \chi_p).
In this paper, we compute the tangent line of some classical Eisenstein series at the `symmetric' center explicitly, and use the result to study the central derivative of certain Hecke L-series.
Local densities of quadratic forms was first introduced by Siegel to study the problem to represent a number (or a positive definite quadratic form) by a (another) positive definite integral form. Local densities are very closely related to local Whittaker functions. In fact, they are two languages to express the same thing, the local factors in the Fourier coefficients of an Eisenstein series. It is very important to have explicit formulas for them in studying arithmetics of Eisenstein series or quadratic forms. However, very few is known except for the `unramified' case (even in the unramified cases, not much is known). The usual method is to use some kind of reduction formula and functional equation to reduce it simple cases one can handle. This kind of method works very poorly in `ramified case', where the reduction formula does not work until in later stage. In this two preprints, we instead compute the local Whittaker functions directly, and have succeeded in obtaining an explicit formula in a quite general situation. In the process, we have also completely determined a1. Gauss integral over $GL_2(\Bbb Z_p)$. In examples related to quaternions, we obtained very nice formulas, which will be used by Kudla and Rapoport to prove an important local identity at a bad prime relating the central derivative of Whittaker functions with local intersection number of cycles in certain Shimura curves.
The following are my graduate work (toward my MS degree) in ring and modular theory, and holomorphic algebra. They are in good quality and indicate my solid algebra background.