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We consider the Laplacian associated with a general metric in the canonicalconformal structure of the noncommutative two torus, and calculate a localexpression for the term a_4 that appears in its corresponding small-time heatkernel expansion. The final formula involves one variable functions and lengthytwo, three and four variable functions of the modular automorphism of the statethat encodes the conformal perturbation of the flat metric. We confirm thevalidity of the calculated expressions by showing that they satisfy a family ofconceptually predicted functional relations. By studying these functionalrelations abstractly, we derive a partial differential system which involves anatural action of cyclic groups of order two, three and four and a flow inparameter space. We discover symmetries of the calculated expressions withrespect to the action of the cyclic groups. In passing, we show that the mainingredients of our calculations, which come from a rearrangement lemma andrelations between the derivatives up to order four of the conformal factor andthose of its logarithm, can be derived by finite differences from thegenerating function of the Bernoulli numbers and its multiplicative inverse. Wethen shed light on the significance of exponential polynomials and their smoothfractions in understanding the general structure of the noncommutativegeometric invariants appearing in the heat kernel expansion. As an applicationof our results we obtain the a_4 term for noncommutative four tori which splitas products of two tori. These four tori are not conformally flat and the a_4term gives a first hint of the Riemann curvature and the higher dimensionalmodular structure.

The term a_4 in the heat kernel expansion of noncommutative tori