Ian Kiming

Lifts of projective congruence groups

We show that noncongruence subgroups of SL2(Z) projectively equivalent to congruence subgroups are ubiquitous. More precisely, they al- ways exist if the congruence subgroup in question is a principal congruence subgroup ( N) of level N > 2, and they exist in many cases also for 0(N). The motivation for asking this question is related to modular forms: pro- jectively equivalent groups have the same spaces of cusp forms for all even weights whereas the spaces of cusp forms of odd weights are distinct in gen- eral. We make some initial observations on this phenomenon for weight 3 via geometric considerations of the attached elliptic modular surfaces. We also develop algorithms that construct all subgroups projectively equiv- alent to a given congruence subgroup and decide which of them are congruence. A crucial tool in this is the generalized level concept of Wohlfahrt.

Mod pq Galois representations and Serre's conjecture

Abstract We consider linear representations of the Galois groups of number fields in two different characteristics and examine conditions under which they arise simultaneously from a motive.

On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials

For an imaginary quadratic fieldK we study the asymptotic behaviour (with respect top) of the number of integers inK with norm of the formk(p−k) for some 1≤k≤p−1, wherep is a prime number. The motivation for studying this problem is that it is known by recent results due to G. Frey and E. Kani that knowledge of this asymptotic behaviour can lead to statements of existence of curves of genus 2 with elliptic differentials in particular cases.We give a general, and from one point of view complete, answer to this question on asymptotic behaviour. This answer is derived from a theorem concerning the number of representations of a natural number by certain quaternary quadratic forms. This second result may be of some independent interest because it can be seen as a generalisation of the classical theorem of Jacobi on the number of representations of a natural number as a sum of 4 squares.

On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N , m, and prime p with p not dividing N , there is only a finite number of characters arising from reductions modulo pm of p-adic representations attached to eigenforms on Γ1(N). We consider various variants of our basic finiteness conjecture, prove some results that support it, and give some numerical evidence.

ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

We consider the classical theta operator

Quadratic Twists of Rigid Calabi–Yau Threefolds Over ℚ

\theta

Lifts of projective congruence groups, II

on modular forms modulo

Congruences like Ramanujan's for powers of the partition function

p^m

On Artin's conjecture for odd 2-dimensional representations

and level

ON MODULAR GALOIS REPRESENTATIONS MODULO PRIME POWERS

N