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On the non-triviality of arithmetic invariants modulo p


Arithmetic invariants are often naturally associated to motives over number fields.

One of the basic questions is the non-triviality of the invariants.

One typically expects generic non-triviality of the invariants as the motive varies in a family.

For a prime $p$, the invariants can often be normalised to be $p$-integral.

One can thus further ask for the generic non-triviality of the invariants modulo $p$.

The invariants can often be expressed in terms of modular forms.

Accordingly, one can try to recast the non-triviality as a modular phenomenon.

If the phenomena can be proven, the non-triviality typically follows in turn.

This principle can be found in the work of Hida and Vatsal among a few others.\


We have been trying to explore a strategy initiated by Hida in the case of central criticial Hecke L-values over the $\Z_p$-anticyclotomic extension of a CM-field.

The strategy crucially relies on a linear indepedence of mod $p$ Hilbert modular forms.

Several arithmetic invariants seem to admit modular expression analogous to the case of Hecke L-values.

This includes the case of Katz $p$-adic L-function, its cyclotomic derivative and $p$-adic Abel-Jacobi image of generalised Heegner cycles.

We approach the non-triviality of these invariants based on the independence.

An analysis of the zero set of the invariants suggests finer versions of the independence.

We approach the versions based on Chai's theory of Hecke stable subvarieties of a mod $p$ Shimura variety.

We formulate a conjecture regarding the analogue of the independence for mod $p$ modular forms on other Shimura varieties.

We prove the analogue in the case of quaternionic Shimura varieties over a totally real field.

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