# Point count statistics for families of curves over finite fields

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A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families.

# Ihara's lemma and local Langlands in families

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In their attempt to mimic the proof of Fermat's Last Theorem for GL(n), Clozel, Harris, and Taylor, were led to a conjectural analogue of Ihara's lemma -- which is still open for n>2. In this talk we will revisit their conjecture from a more modern point of view, and reformulate it in terms of local Langlands in families, as currently being developed by Emerton and Helm. At the end, we hope to hint at potential applications.

# The rank one abelian Gross-Stark conjecture

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Let $\chi$ be a totally odd character of a totally real number field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of Stark which expresses the leading term at s=0 of the p-adic L-function attached to $\chi\omega$ as a product of a regulator and an algebraic number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the rank one case under two assumptions: that Leopoldt's conjecture holds for F and p, and a certain technical condition when there is a unique prime above p in F. After giving some background and outlining their proof, I will explain how to remove both conditions, thus giving an unconditional proof of the conjecture. If there is extra time I will explain an application to the Iwasawa Main Conjecture which comes out of the proof, and make a few remarks on the higher rank case.

# On the Mumford-Tate conjecture for abelian foufolds

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The Mumford-Tate conjecture is a deep conjecture which relates the arithmetic and the geometry of abelian varieties defined over number fields. The results of Moonen and Zarhin indicate that this conjecture holds for almost all absolutely simple abelian fourfolds. The only exception is when the abelian varieties have no nontrivial endomorphism. In this talk we will begin with an introduction to the Mumford-Tate conjecture and a brief summary of known results towards it. Then we sketch a proof of this conjecture in the above 'missing' case for abelian fourfolds.

# Exterior powers of Lubin-Tate groups

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After defining exterior powers of \pi-divisible modules, we prove that the exterior powers of \pi-divisible modules of dimension at most one over any base scheme exist and their construction commute with arbitrary base change

# Moments of Kloosterman sums and modular forms

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Kloosterman sum is one of the most famous exponential sums

in number theory. It is defined using a prime p (and another number).

How do these sums vary with p? Ron Evans has made several conjectures

relating the moment of Kloosterman sums for p to the p-th Fourier

coefficient of certain modular forms. We sketch a proof of his

conjectures.

# A p-adic formula for Chow-Heegner Points

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The Gross-Kudla-Schoen modified diagonal cycle on the triple product of the modular curve with itself provides a wealth of arithmetic information about modular forms, including derivatives of complex L-functions, special values of p-adic L-functions, and points on elliptic curves, known as Chow-Heegner points. In this talk, I will discuss a formula expressing the p-adic logarithm of a Chow-Heegner point in terms of the coefficients of the ordinary projection of certain p-adic modular forms.

# Anticyclotomic Iwasawa invariants and congruences of modular forms

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In this talk, we will look at how congruences between Hecke eigensystems of modular forms affect the Iwasawa invariants of their anticyclotomic p-adic L-functions. It can be regarded as an application of Greenberg-Vatsal's idea on the variation of Iwasawa invariants to the anticyclotomic setting. As an application, we obtain examples of the anticyclotomic main conjecture for modular forms not treated by Skinner-Urban's work. An explicit example will be given.

# Heegner points and a B-SD conjecture

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We prove a B-SD conjecture for elliptic curves (for the p^infinity Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes p. This is done, among other things, by proving a conjecture of Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields. Some applications follow, for example, the p-part of the refined B-SD conjecture in the rank one case.